Theorem nfa2 2177

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2160	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2152	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1851	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1535  Ⅎwnf 1781

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-10 2141  ax-11 2158

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

This theorem is referenced by:  cbv1h  2413  nfra2w  3305  csbie2t  3962  copsex2t  5512  fnoprabg  7573  bj-hbext  36676  bj-nfext  36678  bj-cbv1hv  36762  ax11-pm  36798  pm14.123b  44395  hbexg  44527  nfich2  47322  ich2al  47341