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Theorem bj-rexcom4bv 34269
Description: Version of rexcom4b 3511 and bj-rexcom4b 34270 with a disjoint variable condition on 𝑥, 𝑉, hence removing dependency on df-sb 2071 and df-clab 2803 (so that it depends on df-clel 2896 and df-rex 3139 only on top of first-order logic). Prefer its use over bj-rexcom4b 34270 when sufficient (in particular when 𝑉 is substituted for V). Note the 𝑉 in the hypothesis instead of 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 rexcom4a 3246 . 2 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2 bj-rexcom4bv.1 . . . . 5 𝐵𝑉
32bj-issetiv 34264 . . . 4 𝑥 𝑥 = 𝐵
43biantru 533 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
54rexbii 3242 . 2 (∃𝑦𝐴 𝜑 ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
61, 5bitr4i 281 1 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  wrex 3134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-11 2162
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2896  df-rex 3139
This theorem is referenced by: (None)
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