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Theorem rabeqdv 2680
Description: Equality of restricted class abstractions. Deduction form of rabeq 2678. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypothesis
Ref Expression
rabeqdv.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
rabeqdv  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ps } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem rabeqdv
StepHypRef Expression
1 rabeqdv.1 . 2  |-  ( ph  ->  A  =  B )
2 rabeq 2678 . 2  |-  ( A  =  B  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ps } )
31, 2syl 14 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   {crab 2420
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425
This theorem is referenced by:  dfphi2  11903  cncfval  12738  reldvg  12827  dvfvalap  12829
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