MHB Help with Logic Proof: Establishing 7.

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The discussion focuses on the logic proof from Copi's "Symbolic Logic," specifically how line 7 is established. The user is confused about how the conclusion ¬(∀x) Fx is derived from the earlier lines, particularly given the apparent contradiction between the assumptions. The key point is that lines 1 and 3 cannot both be true simultaneously; thus, if ¬Fy is true (line 2), it leads to the conclusion that not all x can satisfy Fx. The explanation hinges on understanding that the existence of an x for which Fx is false (line 1) contradicts the assertion that Fx holds for all x (line 3), allowing the derivation of line 7. This clarification emphasizes the interplay between existential and universal quantifiers in logical proofs.
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Consider the following proof from Copi's "Symbolic Logic", p. 109:

1. (∃x) ¬Fx Assumption

2. ¬Fy Assumption

3. (∀x) Fx Assumption

4. Fy 3, UI

5. (∀x) Fx ⇒ Fy 3-4, CP

6. ¬(∀x) Fx 5,2 MT

7. ¬(∀x) Fx 1,2-6, EI

8. (∃x) ¬Fx ⇒ ¬(∀x) Fx 1-7, CP

9. (∀x) Fx ⇒ ¬(∃x) ¬Fx 8, Trans, DN

I cannot understand how 7.- is established, everything else is clear. Can someone explain how 1,2-6, EI results in 7. ?
 
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Don't (1), "There exist x such that Fx is not true", and (3), "for all x Fx is true", contradict one another?
 

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