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Theorem riotaeqdv 6268
Description: Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotaeqdv.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
riotaeqdv  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ps ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)    B( x)

Proof of Theorem riotaeqdv
StepHypRef Expression
1 riotaeqdv.1 . . . . 5  |-  ( ph  ->  A  =  B )
21eleq2d 2499 . . . 4  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi1d 709 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ps ) ) )
43iotabidv 5586 . 2  |-  ( ph  ->  ( iota x ( x  e.  A  /\  ps ) )  =  ( iota x ( x  e.  B  /\  ps ) ) )
5 df-riota 6267 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
6 df-riota 6267 . 2  |-  ( iota_ x  e.  B  ps )  =  ( iota x
( x  e.  B  /\  ps ) )
74, 5, 63eqtr4g 2495 1  |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   iotacio 5563   iota_crio 6266
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-rex 2788  df-uni 4223  df-iota 5565  df-riota 6267
This theorem is referenced by:  riotaeqbidv  6270  grpinvpropd  16680  funtransport  30583  fvtransport  30584
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