Theorem ax11-pm 36798

Description: Proof of ax-11 2158 similar to PM's proof of alcom 2160 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 36802. Axiom ax-11 2158 is used in the proof only through nfa2 2177. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax11-pm	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Proof of Theorem ax11-pm

Step	Hyp	Ref	Expression
1		2sp 2187	. . 3 ⊢ (∀𝑥∀𝑦𝜑 → 𝜑)
2	1	gen2 1794	. 2 ⊢ ∀𝑦∀𝑥(∀𝑥∀𝑦𝜑 → 𝜑)
3		nfa2 2177	. . 3 ⊢ Ⅎ𝑦∀𝑥∀𝑦𝜑
4		nfa1 2152	. . 3 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
5	3, 4	2stdpc5 36795	. 2 ⊢ (∀𝑦∀𝑥(∀𝑥∀𝑦𝜑 → 𝜑) → (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
6	2, 5	ax-mp 5	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-or 847  df-ex 1778  df-nf 1782

This theorem is referenced by: (None)