ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabeqif Unicode version

Theorem rabeqif 2740
Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 2739. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
rabeqf.1  |-  F/_ x A
rabeqf.2  |-  F/_ x B
rabeqif.3  |-  A  =  B
Assertion
Ref Expression
rabeqif  |-  { x  e.  A  |  ph }  =  { x  e.  B  |  ph }

Proof of Theorem rabeqif
StepHypRef Expression
1 rabeqif.3 . 2  |-  A  =  B
2 rabeqf.1 . . 3  |-  F/_ x A
3 rabeqf.2 . . 3  |-  F/_ x B
42, 3rabeqf 2739 . 2  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
51, 4ax-mp 5 1  |-  { x  e.  A  |  ph }  =  { x  e.  B  |  ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1363   F/_wnfc 2316   {crab 2469
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474
This theorem is referenced by:  rabeqi  2742
  Copyright terms: Public domain W3C validator