Theorem nfa2 2181

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2182. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2164	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2156	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1854	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1539  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-10 2146  ax-11 2162

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1781  df-nf 1785

This theorem is referenced by:  cbv1h  2407  nfra2w  3269  csbie2t  3884  copsex2t  5435  fnoprabg  7475  bj-hbext  36775  bj-nfext  36777  bj-cbv1hv  36861  ax11-pm  36897  pm14.123b  44543  hbexg  44673  nfich2  47572  ich2al  47591