Theorem nfa2 2171

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2172. (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 1856	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1540  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-10 2138  ax-11 2155

This theorem depends on definitions:  df-bi 206  df-or 847  df-ex 1783  df-nf 1787

This theorem is referenced by:  cbv1h  2405  nfra2w  3297  csbie2t  3935  copsex2t  5493  fnoprabg  7531  bj-hbext  35588  bj-nfext  35590  bj-cbv1hv  35674  ax11-pm  35710  pm14.123b  43185  hbexg  43317  nfich2  46116  ich2al  46135