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Theorem rexcomOLD 3312
Description: Obsolete version of rexcom 3311 as of 26-Aug-2023. Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rexcomOLD (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem rexcomOLD
StepHypRef Expression
1 ancom 464 . . . . 5 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
21anbi1i 626 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜑))
322exbii 1850 . . 3 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑))
4 excom 2167 . . 3 (∃𝑥𝑦((𝑦𝐵𝑥𝐴) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
53, 4bitri 278 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
6 r2ex 3265 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
7 r2ex 3265 . 2 (∃𝑦𝐵𝑥𝐴 𝜑 ↔ ∃𝑦𝑥((𝑦𝐵𝑥𝐴) ∧ 𝜑))
85, 6, 73bitr4i 306 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1781  wcel 2112  wrex 3110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-11 2159
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3114  df-rex 3115
This theorem is referenced by: (None)
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