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Theorem rexcomf 2619
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 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 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 264 . . . . 5 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
21anbi1i 454 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜑))
322exbii 1586 . . 3 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑))
4 excom 1644 . . 3 (∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
53, 4bitri 183 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
6 ralcomf.1 . . 3 𝑦𝐴
76r2exf 2475 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
8 ralcomf.2 . . 3 𝑥𝐵
98r2exf 2475 . 2 (∃𝑦𝐵𝑥𝐴 𝜑 ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
105, 7, 93bitr4i 211 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1472  wcel 2128  wnfc 2286  wrex 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441
This theorem is referenced by:  rexcom  2621
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