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Theorem rexcomf 3091
Description: Commutation of restricted existential 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 466 . . . . 5 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
21anbi1i 730 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜑))
322exbii 1772 . . 3 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑))
4 excom 2039 . . 3 (∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
53, 4bitri 264 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
6 ralcomf.1 . . 3 𝑦𝐴
76r2exf 3055 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
8 ralcomf.2 . . 3 𝑥𝐵
98r2exf 3055 . 2 (∃𝑦𝐵𝑥𝐴 𝜑 ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
105, 7, 93bitr4i 292 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701  wcel 1987  wnfc 2748  wrex 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914
This theorem is referenced by:  rexcom  3093  rexcom4f  29205
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