Axiom ax-7 1441

Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. 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. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ax-7	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Detailed syntax breakdown of Axiom ax-7

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff 𝜑
2		vy	. . . 4 setvar 𝑦
3	1, 2	wal 1346	. . 3 wff ∀𝑦𝜑
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1346	. 2 wff ∀𝑥∀𝑦𝜑
6	1, 4	wal 1346	. . 3 wff ∀𝑥𝜑
7	6, 2	wal 1346	. 2 wff ∀𝑦∀𝑥𝜑
8	5, 7	wi 4	1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This axiom is referenced by:  a7s  1447  alcoms  1469  hbal  1470  alcom  1471  hbald  1484  nfal  1569  hbae  1711  bj-hbalt  13798