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Theorem rspcedeq2vd 3589
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3585 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 2739 . . 3 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐶)
43eqeq2d 2744 . 2 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐷𝐶 = 𝐶))
5 eqidd 2734 . 2 (𝜑𝐶 = 𝐶)
61, 4, 5rspcedvd 3585 1 (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071
This theorem is referenced by:  symgextfo  19212  smatvscl  21896  eucrctshift  29236  ntrclsneine0lem  42428  mogoldbblem  46002  sbgoldbwt  46059  sbgoldbo  46069
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