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Theorem rspcedeq2vd 3458
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3456 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 2766 . . 3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐶)
43eqeq2d 2770 . 2 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐷𝐶 = 𝐶))
5 eqidd 2761 . 2 (𝜑𝐶 = 𝐶)
61, 4, 5rspcedvd 3456 1 (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342
This theorem is referenced by:  symgextfo  18062  smatvscl  20552  eucrctshift  27416  ntrclsneine0lem  38882  mogoldbblem  42157  sbgoldbwt  42193  sbgoldbo  42203
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