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Theorem rexcomf 2489
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 257 . . . . 5 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
21anbi1i 439 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜑))
322exbii 1513 . . 3 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑))
4 excom 1570 . . 3 (∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
53, 4bitri 177 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
6 ralcomf.1 . . 3 𝑦𝐴
76r2exf 2359 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
8 ralcomf.2 . . 3 𝑥𝐵
98r2exf 2359 . 2 (∃𝑦𝐵𝑥𝐴 𝜑 ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
105, 7, 93bitr4i 205 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397  wcel 1409  wnfc 2181  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329
This theorem is referenced by:  rexcom  2491
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