Axiom ax-7 960

Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the 4 axioms of pure predicate calculus. 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. An alternate axiomatization could use ax467 1021 in place of ax-4 971, ax-6o 976, and ax-7 960.

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 set y
3	1, 2	wal 952	. . 3 wff ∀yφ
4		vx	. . 3 set x
5	3, 4	wal 952	. 2 wff ∀x∀yφ
6	1, 4	wal 952	. . 3 wff ∀xφ
7	6, 2	wal 952	. 2 wff ∀y∀xφ
8	5, 7	wi 3	1 wff (∀x∀yφ → ∀y∀xφ)

This axiom is referenced by:  a7s 989  hbal 1003  ax67 1018  ax467 1021  alcom 1030  hbald 1111  hbae 1143  cbv1 1160  sbal1 1344  ax11indalem 1366  ax11inda2ALT 1367  hbabd 1466

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