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 2159	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2151	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1849	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1531  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-10 2141  ax-11 2157

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781

This theorem is referenced by:  cbv1h  2421  csbie2t  3920  copsex2t  5382  fnoprabg  7274  bj-hbext  34044  bj-nfext  34046  bj-cbv1hv  34118  ax11-pm  34155  pm14.123b  40756  hbexg  40888  nfich2  43607  dfich2bi  43614  ich2al  43627