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  hbsbwOLD  2178  nfald  2333  hbae  2435  hbaltg  35987  bj-hbald  36976  bj-nnflemaa  37083  bj-nfald  37449  hbae-o  39349  axc711  39360  axc5c711  39364  ax12indalem  39391  ax12inda2ALT  39392  pm11.71  44824  axc5c4c711  44828  axc11next  44833  hbalg  44982  hbalgVD  45331  hbexgVD  45332  ichal  47926