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Theorem rspcime 2820
 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 108 . . 3 ((𝜑𝑥 = 𝐴) → 𝜑)
42, 32thd 174 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜑))
5 id 19 . 2 (𝜑𝜑)
61, 4, 5rspcedvd 2819 1 (𝜑 → ∃𝑥𝐵 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125  ∃wrex 2433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-v 2711 This theorem is referenced by:  elrnmptdv  4833
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