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Theorem rspcedeq1vd 2772
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2769 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
Hypotheses
Ref Expression
rspcedeqvd.1  |-  ( ph  ->  A  e.  B )
rspcedeqvd.2  |-  ( (
ph  /\  x  =  A )  ->  C  =  D )
Assertion
Ref Expression
rspcedeq1vd  |-  ( ph  ->  E. x  e.  B  C  =  D )
Distinct variable groups:    x, A    x, B    ph, x    x, D
Allowed substitution hint:    C( x)

Proof of Theorem rspcedeq1vd
StepHypRef Expression
1 rspcedeqvd.1 . 2  |-  ( ph  ->  A  e.  B )
2 rspcedeqvd.2 . . 3  |-  ( (
ph  /\  x  =  A )  ->  C  =  D )
32eqeq1d 2126 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( C  =  D  <->  D  =  D ) )
4 eqidd 2118 . 2  |-  ( ph  ->  D  =  D )
51, 3, 4rspcedvd 2769 1  |-  ( ph  ->  E. x  e.  B  C  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   E.wrex 2394
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662
This theorem is referenced by: (None)
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