Axiom ax-11 2157

Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 2129) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)

Assertion

Ref	Expression
ax-11	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Detailed syntax breakdown of Axiom ax-11

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff 𝜑
2		vy	. . . 4 setvar 𝑦
3	1, 2	wal 1539	. . 3 wff ∀𝑦𝜑
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1539	. 2 wff ∀𝑥∀𝑦𝜑
6	1, 4	wal 1539	. . 3 wff ∀𝑥𝜑
7	6, 2	wal 1539	. 2 wff ∀𝑦∀𝑥𝜑
8	5, 7	wi 4	1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This axiom is referenced by:  alcoms  2158  alcom  2159  hbal  2170  hbald  2171  hbsbw  2172  nfald  2325  hbae  2432  hbaltg  33762  bj-hbalt  34842  bj-nnflemaa  34923  bj-nfald  35287  hbae-o  36896  axc711  36907  axc5c711  36911  ax12indalem  36938  ax12inda2ALT  36939  pm11.71  41968  axc5c4c711  41972  axc11next  41977  hbalg  42128  hbalgVD  42478  hbexgVD  42479  ichal  44870