Theorem nfa2 2213

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2216. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2197	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2189	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1886	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

This proof depends on syntax axioms:  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-10 2179  ax-11 2195

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  cbv1h  2440  nfra2w  3304  csbie2t  3894  copsex2t  5480  fnoprabg  7546  bj-nfext  37380  bj-cbv1hv  37472  ax11-pm  37508  pm14.123b  45177  hbexg  45306  nfich2  48238  ich2al  48257