Theorem nfa2 2208

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2211. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2192	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2184	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1872	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1557  Ⅎwnf 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-10 2174  ax-11 2190

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1799  df-nf 1803

This theorem is referenced by:  cbv1h  2435  nfra2w  3297  csbie2t  3890  copsex2t  5460  fnoprabg  7515  bj-nfext  37153  bj-cbv1hv  37245  ax11-pm  37281  pm14.123b  44966  hbexg  45096  nfich2  48018  ich2al  48037