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 2163	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2155	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1853	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Syntax hints:  ∀wal 1535  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 2145  ax-11 2161

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by:  cbv1h  2425  csbie2t  3921  copsex2t  5383  fnoprabg  7275  bj-hbext  34044  bj-nfext  34046  bj-cbv1hv  34118  ax11-pm  34155  pm14.123b  40778  hbexg  40910  nfich2  43628  dfich2bi  43635  ich2al  43648