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Theorem rspceeqv 2859
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 2189 . 2 (𝑥 = 𝐴 → (𝐸 = 𝐶𝐸 = 𝐷))
32rspcev 2841 1 ((𝐴𝐵𝐸 = 𝐷) → ∃𝑥𝐵 𝐸 = 𝐶)
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:  elixpsn  6728  ixpsnf1o  6729  elfir  6965  0ct  7099  ctmlemr  7100  ctssdclemn0  7102  fodju0  7138  mertenslemi1  11514  mertenslem2  11515  pcprmpw  12303  1arithlem4  12334  ctiunctlemfo  12410  elrestr  12631  restopnb  13314  mopnex  13638  metrest  13639  2sqlem2  14084  mul2sq  14085  2sqlem3  14086  2sqlem9  14093  2sqlem10  14094
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