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Theorem rexcomf 2770
 Description: Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
ralcomf.1 yA
ralcomf.2 xB
Assertion
Ref Expression
rexcomf (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 rexcomf
StepHypRef Expression
1 ancom 437 . . . . 5 ((x A y B) ↔ (y B x A))
21anbi1i 676 . . . 4 (((x A y B) φ) ↔ ((y B x A) φ))
322exbii 1583 . . 3 (xy((x A y B) φ) ↔ xy((y B x A) φ))
4 excom 1741 . . 3 (xy((y B x A) φ) ↔ yx((y B x A) φ))
53, 4bitri 240 . 2 (xy((x A y B) φ) ↔ yx((y B x A) φ))
6 ralcomf.1 . . 3 yA
76r2exf 2650 . 2 (x A y B φxy((x A y B) φ))
8 ralcomf.2 . . 3 xB
98r2exf 2650 . 2 (y B x A φyx((y B x A) φ))
105, 7, 93bitr4i 268 1 (x A y B φy B x A φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  Ⅎwnfc 2476  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-rex 2620 This theorem is referenced by:  rexcom  2772
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