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Theorem rspcedeq1vd 2681
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2680 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 2064 . 2 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐷𝐷 = 𝐷))
4 eqidd 2057 . 2 (𝜑𝐷 = 𝐷)
51, 3, 4rspcedvd 2680 1 (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576
This theorem is referenced by: (None)
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