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  35397  bj-hbalt  36152  bj-nnflemaa  36233  bj-nfald  36610  hbae-o  38369  axc711  38380  axc5c711  38384  ax12indalem  38411  ax12inda2ALT  38412  pm11.71  43828  axc5c4c711  43832  axc11next  43837  hbalg  43988  hbalgVD  44338  hbexgVD  44339  ichal  46800