Theorem nfa2 2165

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2166. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2148	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2140	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1847	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1531  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-10 2129  ax-11 2146

This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1774  df-nf 1778

This theorem is referenced by:  cbv1h  2398  nfra2w  3286  csbie2t  3930  copsex2t  5494  fnoprabg  7543  bj-hbext  36318  bj-nfext  36320  bj-cbv1hv  36404  ax11-pm  36440  pm14.123b  44005  hbexg  44137  nfich2  46925  ich2al  46944