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Theorem rexcomOLD 3273
Description: Obsolete version of rexcom 3272 as of 8-Dec-2024. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof shortened by BJ, 26-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
rexcomOLD (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem rexcomOLD
StepHypRef Expression
1 df-rex 3071 . . 3 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
21rexbii 3094 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦(𝑦𝐵𝜑))
3 rexcom4 3270 . 2 (∃𝑥𝐴𝑦(𝑦𝐵𝜑) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵𝜑))
4 r19.42v 3184 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
54exbii 1851 . . 3 (∃𝑦𝑥𝐴 (𝑦𝐵𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
6 df-rex 3071 . . 3 (∃𝑦𝐵𝑥𝐴 𝜑 ↔ ∃𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
75, 6bitr4i 278 . 2 (∃𝑦𝑥𝐴 (𝑦𝐵𝜑) ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
82, 3, 73bitri 297 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1782  wcel 2107  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-11 2155
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-rex 3071
This theorem is referenced by: (None)
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