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Theorem rspcedeq2vd 3314
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3312 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 2626 . . 3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐶)
43eqeq2d 2630 . 2 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐷𝐶 = 𝐶))
5 eqidd 2621 . 2 (𝜑𝐶 = 𝐶)
61, 4, 5rspcedvd 3312 1 (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wrex 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197
This theorem is referenced by:  symgextfo  17823  smatvscl  20311  eucrctshift  27083  ntrclsneine0lem  38182  mogoldbblem  41394  sbgoldbwt  41430  sbgoldbo  41440
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