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Theorem bj-rexcom4b 32997
 Description: Remove from rexcom4b 3258 dependency on ax-ext 2631 and ax-13 2282 (and on df-or 384, df-cleq 2644, df-nfc 2782, df-v 3233). The hypothesis uses 𝑉 instead of V (see bj-isseti 32989 for the motivation). Use bj-rexcom4bv 32996 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 bj-rexcom4a 32995 . 2 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2 bj-rexcom4b.1 . . . . 5 𝐵𝑉
32bj-isseti 32989 . . . 4 𝑥 𝑥 = 𝐵
43biantru 525 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
54rexbii 3070 . 2 (∃𝑦𝐴 𝜑 ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
61, 5bitr4i 267 1 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃wrex 2942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-11 2074  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938  df-clab 2638  df-clel 2647  df-rex 2947 This theorem is referenced by: (None)
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