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Theorem bj-rexcom4a 33097
Description: Remove from rexcom4a 3330 dependency on ax-ext 2704 and ax-13 2355 (and on df-or 384, df-sb 2011, df-clab 2711, df-cleq 2717, df-clel 2720, df-nfc 2855, df-v 3306). This proof uses only df-rex 3020 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-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 bj-rexcom4a
StepHypRef Expression
1 bj-rexcom4 33096 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑥𝑦𝐴 (𝜑𝜓))
2 19.42v 1994 . . 3 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
32rexbii 3143 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
41, 3bitr3i 266 1 (∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1817  wrex 3015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-11 2147
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1818  df-rex 3020
This theorem is referenced by:  bj-rexcom4bv  33098  bj-rexcom4b  33099
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