Axiom ax-11 2021

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 1994) 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 1473	. . 3 wff ∀𝑦𝜑
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1473	. 2 wff ∀𝑥∀𝑦𝜑
6	1, 4	wal 1473	. . 3 wff ∀𝑥𝜑
7	6, 2	wal 1473	. 2 wff ∀𝑦∀𝑥𝜑
8	5, 7	wi 4	1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This axiom is referenced by:  alcoms  2022  hbal  2023  alcom  2024  hbald  2028  nfald  2151  hbae  2303  hbaltg  30957  bj-hbalt  31858  hbae-o  33206  axc711  33217  axc5c711  33221  ax12indalem  33248  ax12inda2ALT  33249  pm11.71  37619  axc5c4c711  37624  axc11next  37629  hbalg  37792  hbalgVD  38163  hbexgVD  38164