MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabeqf Structured version   Unicode version

Theorem rabeqf 3080
Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
Hypotheses
Ref Expression
rabeqf.1  |-  F/_ x A
rabeqf.2  |-  F/_ x B
Assertion
Ref Expression
rabeqf  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )

Proof of Theorem rabeqf
StepHypRef Expression
1 rabeqf.1 . . . 4  |-  F/_ x A
2 rabeqf.2 . . . 4  |-  F/_ x B
31, 2nfeq 2602 . . 3  |-  F/ x  A  =  B
4 eleq2 2502 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 709 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5abbid 2564 . 2  |-  ( A  =  B  ->  { x  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  B  /\  ph ) } )
7 df-rab 2791 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
8 df-rab 2791 . 2  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
96, 7, 83eqtr4g 2495 1  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {cab 2414   F/_wnfc 2577   {crab 2786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791
This theorem is referenced by:  rabeq  3081  fpwrelmapffs  28162  rabeq12f  32104
  Copyright terms: Public domain W3C validator