Axiom ax-11 2163

Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. 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 ax11w 2136) but is used as an auxiliary axiom scheme to achieve metalogical completeness. Use its weak version alcomimw 2045 when it allows to avoid dependence on ax-11 2163. (Contributed by NM, 12-Mar-1993.)

Assertion

Ref	Expression
ax-11	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Detailed syntax breakdown of Axiom ax-11

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:  alcoms  2164  alcom  2165  hbal  2173  hbald  2174  nfald  2332  hbae  2434  hbaltg  35975  bj-hbald  36964  bj-nnflemaa  37071  bj-nfald  37437  hbae-o  39337  axc711  39348  axc5c711  39352  ax12indalem  39379  ax12inda2ALT  39380  pm11.71  44812  axc5c4c711  44816  axc11next  44821  hbalg  44970  hbalgVD  45319  hbexgVD  45320  ichal  47914