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Theorem ralcomf 2769
 Description: Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
ralcomf.1 yA
ralcomf.2 xB
Assertion
Ref Expression
ralcomf (x A y B φy B x A φ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   A(x,y)   B(x,y)

Proof of Theorem ralcomf
StepHypRef Expression
1 ancomsimp 1369 . . . 4 (((x A y B) → φ) ↔ ((y B x A) → φ))
212albii 1567 . . 3 (xy((x A y B) → φ) ↔ xy((y B x A) → φ))
3 alcom 1737 . . 3 (xy((y B x A) → φ) ↔ yx((y B x A) → φ))
42, 3bitri 240 . 2 (xy((x A y B) → φ) ↔ yx((y B x A) → φ))
5 ralcomf.1 . . 3 yA
65r2alf 2649 . 2 (x A y B φxy((x A y B) → φ))
7 ralcomf.2 . . 3 xB
87r2alf 2649 . 2 (y B x A φyx((y B x A) → φ))
94, 6, 83bitr4i 268 1 (x A y B φy B x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by:  ralcom  2771  ssiinf  4015
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