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Theorem bj-rexcom4bv 32518
Description: Version of bj-rexcom4b 32519 with a dv condition on 𝑥, 𝑉, hence removing dependency on df-sb 1878 and df-clab 2608 (so that it depends on df-clel 2617 and df-rex 2913 only on top of first-order logic). Prefer its use over bj-rexcom4b 32519 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexcom4bv.1 𝐵𝑉
Assertion
Ref Expression
bj-rexcom4bv (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝑉(𝑦)

Proof of Theorem bj-rexcom4bv
StepHypRef Expression
1 bj-rexcom4a 32517 . 2 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2 bj-rexcom4bv.1 . . . . 5 𝐵𝑉
32bj-issetiv 32510 . . . 4 𝑥 𝑥 = 𝐵
43biantru 526 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
54rexbii 3034 . 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 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-11 2031
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-clel 2617  df-rex 2913
This theorem is referenced by: (None)
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