Axiom ax-11 2147

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 2119) but is used as an auxiliary axiom scheme to achieve metalogical completeness. Use its weak version alcomiw 2039 when it allows to avoid dependence on ax-11 2147. (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 1532	. . 3 wff ∀𝑦𝜑
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1532	. 2 wff ∀𝑥∀𝑦𝜑
6	1, 4	wal 1532	. . 3 wff ∀𝑥𝜑
7	6, 2	wal 1532	. 2 wff ∀𝑦∀𝑥𝜑
8	5, 7	wi 4	1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This axiom is referenced by:  alcoms  2148  alcom  2149  hbal  2157  hbald  2158  hbsbwOLD  2162  nfald  2317  hbae  2426  hbaltg  35403  bj-hbalt  36158  bj-nnflemaa  36239  bj-nfald  36616  hbae-o  38375  axc711  38386  axc5c711  38390  ax12indalem  38417  ax12inda2ALT  38418  pm11.71  43834  axc5c4c711  43838  axc11next  43843  hbalg  43994  hbalgVD  44344  hbexgVD  44345  ichal  46806