Axiom ax-7 1734

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. This axiom scheme is logically redundant (see ax7w 1718) but is used as an auxiliary axiom to achieve metalogical completeness. (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 1540	. . 3 wff ∀yφ
4		vx	. . 3 setvar x
5	3, 4	wal 1540	. 2 wff ∀x∀yφ
6	1, 4	wal 1540	. . 3 wff ∀xφ
7	6, 2	wal 1540	. 2 wff ∀y∀xφ
8	5, 7	wi 4	1 wff (∀x∀yφ → ∀y∀xφ)

This axiom is referenced by:  a7s  1735  hbal  1736  alcom  1737  hbald  1740  nfaldOLD  1853  hbae  1953  cbv1h  1978  sbal1  2126  hbae-o  2153  ax67  2165  ax467  2169  ax11indalem  2197  ax11inda2ALT  2198