Theorem nfa2 2212

Description: Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2215. (Revised by Wolf Lammen, 18-Oct-2021.)

Assertion

Ref	Expression
nfa2	⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑

Proof of Theorem nfa2

Step	Hyp	Ref	Expression
1		alcom 2196	. 2 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
2		nfa1 2188	. 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 2178  ax-11 2194

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

This theorem is used by:  cbv1h  2436  nfra2w  3300  csbie2t  3888  copsex2t  5473  fnoprabg  7540  bj-nfext  37455  bj-cbv1hv  37547  ax11-pm  37583  pm14.123b  45258  hbexg  45387  nfich2  48356  ich2al  48375