Theorem nfa2 2172

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2173. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2158	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2150	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1856	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-10 2139  ax-11 2156

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by:  cbv1h  2405  csbie2t  3869  copsex2t  5400  fnoprabg  7375  bj-hbext  34819  bj-nfext  34821  bj-cbv1hv  34905  ax11-pm  34942  pm14.123b  41933  hbexg  42065  nfich2  44788  ich2al  44807