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Theorem rspcedeq2vd 2826
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2822 for equations, in which the right 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
rspcedeq2vd  |-  ( ph  ->  E. x  e.  B  C  =  D )
Distinct variable groups:    x, A    x, B    ph, x    x, C
Allowed substitution hint:    D( x)

Proof of Theorem rspcedeq2vd
StepHypRef Expression
1 rspcedeqvd.1 . 2  |-  ( ph  ->  A  e.  B )
2 rspcedeqvd.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  C  =  D )
32eqcomd 2163 . . 3  |-  ( (
ph  /\  x  =  A )  ->  D  =  C )
43eqeq2d 2169 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( C  =  D  <->  C  =  C ) )
5 eqidd 2158 . 2  |-  ( ph  ->  C  =  C )
61, 4, 5rspcedvd 2822 1  |-  ( ph  ->  E. x  e.  B  C  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   E.wrex 2436
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714
This theorem is referenced by: (None)
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