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Mirrors > Home > ILE Home > Th. List > ax-7 | GIF version |
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.) |
Ref | Expression |
---|---|
ax-7 | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
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) |
Colors of variables: wff set class |
This axiom is referenced by: a7s 1447 alcoms 1469 hbal 1470 alcom 1471 hbald 1484 nfal 1569 hbae 1711 bj-hbalt 13757 |
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