Theorem ax11-pm 37255

Description: Proof of ax-11 2181 similar to PM's proof of alcom 2183 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 37259. Axiom ax-11 2181 is used in the proof only through nfa2 2199. (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 2211	. . 3 ⊢ (∀𝑥∀𝑦𝜑 → 𝜑)
2	1	gen2 1806	. 2 ⊢ ∀𝑦∀𝑥(∀𝑥∀𝑦𝜑 → 𝜑)
3		nfa2 2199	. . 3 ⊢ Ⅎ𝑦∀𝑥∀𝑦𝜑
4		nfa1 2175	. . 3 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
5	3, 4	2stdpc5 37252	. 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 1548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-10 2165  ax-11 2181  ax-12 2202

This theorem depends on definitions:  df-bi 209  df-or 857  df-ex 1790  df-nf 1794

This theorem is referenced by: (None)