Axiom ax-11 2031

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 2004) 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 1478	. . 3 wff ∀𝑦𝜑
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1478	. 2 wff ∀𝑥∀𝑦𝜑
6	1, 4	wal 1478	. . 3 wff ∀𝑥𝜑
7	6, 2	wal 1478	. 2 wff ∀𝑦∀𝑥𝜑
8	5, 7	wi 4	1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This axiom is referenced by:  alcoms  2032  hbal  2033  alcom  2034  hbald  2038  nfald  2162  hbae  2314  hbaltg  31411  bj-hbalt  32310  hbae-o  33665  axc711  33676  axc5c711  33680  ax12indalem  33707  ax12inda2ALT  33708  pm11.71  38076  axc5c4c711  38081  axc11next  38086  hbalg  38250  hbalgVD  38621  hbexgVD  38622