Theorem axc711to11 38859

Description: Rederivation of ax-11 2156 from axc711 38856. Note that ax-c7 38827 and ax-11 2156 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc711to11	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Proof of Theorem axc711to11

Step	Hyp	Ref	Expression
1		axc711toc7 38858	. . 3 ⊢ (¬ ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ¬ ∀𝑥∀𝑦𝜑)
2	1	con4i 114	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑)
3		axc711 38856	. . 3 ⊢ (¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑)
4	3	alimi 1810	. 2 ⊢ (∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
5	2, 4	syl 17	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-11 2156  ax-c5 38825  ax-c4 38826  ax-c7 38827

This theorem is referenced by: (None)