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Theorem rspcime 3578
 Description: Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypotheses
Ref Expression
rspcime.1 ((𝜑𝑥 = 𝐴) → 𝜓)
rspcime.2 (𝜑𝐴𝐵)
Assertion
Ref Expression
rspcime (𝜑 → ∃𝑥𝐵 𝜓)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐵   𝑥,𝐴
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcime
StepHypRef Expression
1 rspcime.2 . 2 (𝜑𝐴𝐵)
2 rspcime.1 . . 3 ((𝜑𝑥 = 𝐴) → 𝜓)
3 simpl 486 . . 3 ((𝜑𝑥 = 𝐴) → 𝜑)
42, 32thd 268 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜑))
5 id 22 . 2 (𝜑𝜑)
61, 4, 5rspcedvd 3577 1 (𝜑 → ∃𝑥𝐵 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∃wrex 3110 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115 This theorem is referenced by:  elrnmptdv  5802  mnuprdlem3  40969  mnurndlem1  40976  grumnudlem  40980  grumnud  40981  inaex  40992  gruex  40993
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