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Theorem bj-rexcom4b 34096
Description: Remove from rexcom4b 3522 dependency on ax-ext 2790 and ax-13 2381 (and on df-or 842, df-cleq 2811, df-nfc 2960, df-v 3494). The hypothesis uses 𝑉 instead of V (see bj-isseti 34091 for the motivation). Use bj-rexcom4bv 34095 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexcom4b.1 𝐵𝑉
Assertion
Ref Expression
bj-rexcom4b (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem bj-rexcom4b
StepHypRef Expression
1 rexcom4a 3248 . 2 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2 bj-rexcom4b.1 . . . . 5 𝐵𝑉
32bj-isseti 34091 . . . 4 𝑥 𝑥 = 𝐵
43biantru 530 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
54rexbii 3244 . 2 (∃𝑦𝐴 𝜑 ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
61, 5bitr4i 279 1 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-11 2151
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-clel 2890  df-rex 3141
This theorem is referenced by: (None)
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