Theorem nfa2 2174

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2175. (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 1854	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1536  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-10 2142  ax-11 2158

This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  cbv1h  2414  csbie2t  3866  copsex2t  5348  fnoprabg  7254  bj-hbext  34157  bj-nfext  34159  bj-cbv1hv  34233  ax11-pm  34270  pm14.123b  41130  hbexg  41262  nfich2  43965  ich2al  43984