Theorem nfa2 2186

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2189. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2170	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2162	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1860	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-10 2152  ax-11 2168

This theorem depends on definitions:  df-bi 208  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbv1h  2413  nfra2w  3276  csbie2t  3876  copsex2t  5440  fnoprabg  7486  bj-nfext  37064  bj-cbv1hv  37156  ax11-pm  37192  pm14.123b  44877  hbexg  45007  nfich2  47930  ich2al  47949