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Theorem rspcedeq1vd 2799
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2796 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
Hypotheses
Ref Expression
rspcedeqvd.1 (𝜑𝐴𝐵)
rspcedeqvd.2 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)
Assertion
Ref Expression
rspcedeq1vd (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝑥,𝐷
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem rspcedeq1vd
StepHypRef Expression
1 rspcedeqvd.1 . 2 (𝜑𝐴𝐵)
2 rspcedeqvd.2 . . 3 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)
32eqeq1d 2149 . 2 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐷𝐷 = 𝐷))
4 eqidd 2141 . 2 (𝜑𝐷 = 𝐷)
51, 3, 4rspcedvd 2796 1 (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  wrex 2418
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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2689
This theorem is referenced by: (None)
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