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Theorem rspcedeq2vd 2711
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2709 for equations, in which the right 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
rspcedeq2vd (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝑥,𝐶
Allowed substitution hint:   𝐷(𝑥)

Proof of Theorem rspcedeq2vd
StepHypRef Expression
1 rspcedeqvd.1 . 2 (𝜑𝐴𝐵)
2 rspcedeqvd.2 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)
32eqcomd 2087 . . 3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐶)
43eqeq2d 2093 . 2 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐷𝐶 = 𝐶))
5 eqidd 2083 . 2 (𝜑𝐶 = 𝐶)
61, 4, 5rspcedvd 2709 1 (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604
This theorem is referenced by: (None)
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