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Theorem ceqsrexv2 33011
Description: Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)
Hypothesis
Ref Expression
ceqsrexv2.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsrexv2 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsrexv2
StepHypRef Expression
1 ceqsrexv2.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
21ceqsrexbv 3636 1 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  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-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896  df-rex 3139
This theorem is referenced by:  ceqsralv2  33013
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