Axiom ax-11 2194

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 2167) but is used as an auxiliary axiom scheme to achieve metalogical completeness. Use its weak version alcomimw 2076 when it allows to avoid dependence on ax-11 2194. (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 1568	. . 3 wff ∀𝑦𝜑
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1568	. 2 wff ∀𝑥∀𝑦𝜑
6	1, 4	wal 1568	. . 3 wff ∀𝑥𝜑
7	6, 2	wal 1568	. 2 wff ∀𝑦∀𝑥𝜑
8	5, 7	wi 4	1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This axiom is used by:  alcoms  2195  alcom  2196  hbal  2204  hbald  2205  nfald  2358  hbae  2460  hbaltg  36491  bj-hbald  37503  bj-nnflemaa  37610  bj-nfald  37976  findcard4  38552  hbae-o  39880  axc711  39891  axc5c711  39895  ax12indalem  39922  ax12inda2ALT  39923  pm11.71  45325  axc5c4c711  45329  axc11next  45334  hbalg  45482  hbalgVD  45831  hbexgVD  45832  ichal  48470