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 2157	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2149	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1850	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1535  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 2139  ax-11 2155

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777  df-nf 1781

This theorem is referenced by:  cbv1h  2408  nfra2w  3297  csbie2t  3949  copsex2t  5503  fnoprabg  7556  bj-hbext  36693  bj-nfext  36695  bj-cbv1hv  36779  ax11-pm  36815  pm14.123b  44422  hbexg  44554  nfich2  47373  ich2al  47392