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Theorem rexcom4a 3217
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
Assertion
Ref Expression
rexcom4a (∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem rexcom4a
StepHypRef Expression
1 rexcom4 3215 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑥𝑦𝐴 (𝜑𝜓))
2 19.42v 1954 . . 3 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
32rexbii 3213 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
41, 3bitr3i 280 1 (∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1781  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-rex 3115
This theorem is referenced by:  rexcom4b  3474  bj-rexcom4bv  34317  bj-rexcom4b  34318
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