Theorem nfa2 2176

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2177. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2159	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2151	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1853	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-10 2141  ax-11 2157

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780  df-nf 1784

This theorem is referenced by:  cbv1h  2410  nfra2w  3299  csbie2t  3937  copsex2t  5497  fnoprabg  7556  bj-hbext  36711  bj-nfext  36713  bj-cbv1hv  36797  ax11-pm  36833  pm14.123b  44445  hbexg  44576  nfich2  47435  ich2al  47454