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  2334  hbae  2436  hbaltg  36027  bj-hbald  36944  bj-nnflemaa  37051  bj-nfald  37417  hbae-o  39308  axc711  39319  axc5c711  39323  ax12indalem  39350  ax12inda2ALT  39351  pm11.71  44782  axc5c4c711  44786  axc11next  44791  hbalg  44940  hbalgVD  45289  hbexgVD  45290  ichal  47855