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Axiom ax-7 1337
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 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Detailed syntax breakdown of Axiom ax-7
StepHypRef Expression
1 wph . . . 4 wff 𝜑
2 vy . . . 4 setvar 𝑦
31, 2wal 1241 . . 3 wff 𝑦𝜑
4 vx . . 3 setvar 𝑥
53, 4wal 1241 . 2 wff 𝑥𝑦𝜑
61, 4wal 1241 . . 3 wff 𝑥𝜑
76, 2wal 1241 . 2 wff 𝑦𝑥𝜑
85, 7wi 4 1 wff (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff set class
This axiom is referenced by:  a7s  1343  alcoms  1365  hbal  1366  alcom  1367  hbald  1380  hbae  1606  bj-hbalt  9750
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