Axiom ax-11 2161

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 2134) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (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 1535	. . 3 wff ∀𝑦𝜑
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1535	. 2 wff ∀𝑥∀𝑦𝜑
6	1, 4	wal 1535	. . 3 wff ∀𝑥𝜑
7	6, 2	wal 1535	. 2 wff ∀𝑦∀𝑥𝜑
8	5, 7	wi 4	1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This axiom is referenced by:  alcoms  2162  alcom  2163  hbal  2174  hbald  2175  nfald  2347  hbae  2453  hbaltg  33052  bj-hbalt  34015  bj-nnflemaa  34091  bj-nfald  34432  hbae-o  36054  axc711  36065  axc5c711  36069  ax12indalem  36096  ax12inda2ALT  36097  pm11.71  40778  axc5c4c711  40782  axc11next  40787  hbalg  40938  hbalgVD  41288  hbexgVD  41289  ichal  43676