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Theorem bj-rexcom4 33191
Description: Remove from rexcom4 3365 dependency on ax-ext 2740 and ax-13 2391 (and on df-or 384, df-tru 1635, df-sb 2047, df-clab 2747, df-cleq 2753, df-clel 2756, df-nfc 2891, df-v 3342). This proof uses only df-rex 3056 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 3056 . 2 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝜑))
2 19.42v 2030 . . . . 5 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
32bicomi 214 . . . 4 ((𝑥𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦(𝑥𝐴𝜑))
43exbii 1923 . . 3 (∃𝑥(𝑥𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑥𝑦(𝑥𝐴𝜑))
5 excom 2191 . . . 4 (∃𝑥𝑦(𝑥𝐴𝜑) ↔ ∃𝑦𝑥(𝑥𝐴𝜑))
6 df-rex 3056 . . . . . 6 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
76bicomi 214 . . . . 5 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥𝐴 𝜑)
87exbii 1923 . . . 4 (∃𝑦𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥𝐴 𝜑)
95, 8bitri 264 . . 3 (∃𝑥𝑦(𝑥𝐴𝜑) ↔ ∃𝑦𝑥𝐴 𝜑)
104, 9bitri 264 . 2 (∃𝑥(𝑥𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦𝑥𝐴 𝜑)
111, 10bitri 264 1 (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1853  wcel 2139  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-11 2183
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-rex 3056
This theorem is referenced by:  bj-rexcom4a  33192
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