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 1853	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-10 2142  ax-11 2158

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1780  df-nf 1784

This theorem is referenced by:  cbv1h  2410  nfra2w  3284  csbie2t  3917  copsex2t  5472  fnoprabg  7535  bj-hbext  36733  bj-nfext  36735  bj-cbv1hv  36819  ax11-pm  36855  pm14.123b  44417  hbexg  44548  nfich2  47429  ich2al  47448