Theorem nfa2 2209

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2212. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2195	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2187	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
3	1, 2	nfxfr 1885	1 ⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

This proof depends on syntax axioms:  ∀wal 1567  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-10 2177  ax-11 2193

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  cbv1h  2434  nfra2w  3298  csbie2t  3884  copsex2t  5463  fnoprabg  7543  bj-nfext  37708  bj-cbv1hv  37800  ax11-pm  37836  pm14.123b  45503  hbexg  45632  nfich2  48623  ich2al  48642