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Theorem rexcomf 2446
 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 253 . . . . 5 ((x A y B) ↔ (y B x A))
21anbi1i 434 . . . 4 (((x A y B) φ) ↔ ((y B x A) φ))
322exbii 1475 . . 3 (xy((x A y B) φ) ↔ xy((y B x A) φ))
4 excom 1532 . . 3 (xy((y B x A) φ) ↔ yx((y B x A) φ))
53, 4bitri 173 . 2 (xy((x A y B) φ) ↔ yx((y B x A) φ))
6 ralcomf.1 . . 3 yA
76r2exf 2316 . 2 (x A y B φxy((x A y B) φ))
8 ralcomf.2 . . 3 xB
98r2exf 2316 . 2 (y B x A φyx((y B x A) φ))
105, 7, 93bitr4i 201 1 (x A y B φy B x A φ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1358   ∈ wcel 1370  Ⅎwnfc 2143  ∃wrex 2281 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286 This theorem is referenced by:  rexcom  2448
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