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Theorem rspcedeq2vd 2851
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2847 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 2183 . . 3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐶)
43eqeq2d 2189 . 2 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐷𝐶 = 𝐶))
5 eqidd 2178 . 2 (𝜑𝐶 = 𝐶)
61, 4, 5rspcedvd 2847 1 (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739
This theorem is referenced by: (None)
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