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Theorem rspceeqv 2812
 Description: Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
Hypothesis
Ref Expression
rspceeqv.1 (𝑥 = 𝐴𝐶 = 𝐷)
Assertion
Ref Expression
rspceeqv ((𝐴𝐵𝐸 = 𝐷) → ∃𝑥𝐵 𝐸 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem rspceeqv
StepHypRef Expression
1 rspceeqv.1 . . 3 (𝑥 = 𝐴𝐶 = 𝐷)
21eqeq2d 2152 . 2 (𝑥 = 𝐴 → (𝐸 = 𝐶𝐸 = 𝐷))
32rspcev 2794 1 ((𝐴𝐵𝐸 = 𝐷) → ∃𝑥𝐵 𝐸 = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ∃wrex 2418 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2692 This theorem is referenced by:  elixpsn  6638  ixpsnf1o  6639  elfir  6871  0ct  7002  ctmlemr  7003  ctssdclemn0  7005  fodju0  7029  mertenslemi1  11356  mertenslem2  11357  ctiunctlemfo  12008  elrestr  12187  restopnb  12409  mopnex  12733  metrest  12734
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