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Theorem bj-rexcom4 32063
 Description: Remove from rexcom4 3198 dependency on ax-ext 2590 and ax-13 2234 (and on df-or 384, df-tru 1478, df-sb 1868, df-clab 2597, df-cleq 2603, df-clel 2606, df-nfc 2740, df-v 3175). This proof uses only df-rex 2902 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-rexcom4 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem bj-rexcom4
StepHypRef Expression
1 df-rex 2902 . 2 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝜑))
2 19.42v 1905 . . . . 5 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
32bicomi 213 . . . 4 ((𝑥𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦(𝑥𝐴𝜑))
43exbii 1764 . . 3 (∃𝑥(𝑥𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑥𝑦(𝑥𝐴𝜑))
5 excom 2029 . . . 4 (∃𝑥𝑦(𝑥𝐴𝜑) ↔ ∃𝑦𝑥(𝑥𝐴𝜑))
6 df-rex 2902 . . . . . 6 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
76bicomi 213 . . . . 5 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥𝐴 𝜑)
87exbii 1764 . . . 4 (∃𝑦𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥𝐴 𝜑)
95, 8bitri 263 . . 3 (∃𝑥𝑦(𝑥𝐴𝜑) ↔ ∃𝑦𝑥𝐴 𝜑)
104, 9bitri 263 . 2 (∃𝑥(𝑥𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦𝑥𝐴 𝜑)
111, 10bitri 263 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-11 2021 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-rex 2902 This theorem is referenced by:  bj-rexcom4a  32064
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