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Theorem riotabiia 6174
Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 3061 analog.) (Contributed by NM, 16-Jan-2012.)
Hypothesis
Ref Expression
riotabiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
riotabiia  |-  ( iota_ x  e.  A  ph )  =  ( iota_ x  e.  A  ps )

Proof of Theorem riotabiia
StepHypRef Expression
1 eqid 2452 . 2  |-  _V  =  _V
2 riotabiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
32adantl 466 . . 3  |-  ( ( _V  =  _V  /\  x  e.  A )  ->  ( ph  <->  ps )
)
43riotabidva 6173 . 2  |-  ( _V  =  _V  ->  ( iota_ x  e.  A  ph )  =  ( iota_ x  e.  A  ps )
)
51, 4ax-mp 5 1  |-  ( iota_ x  e.  A  ph )  =  ( iota_ x  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   _Vcvv 3072   iota_crio 6155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-rex 2802  df-uni 4195  df-iota 5484  df-riota 6156
This theorem is referenced by:  riotaxfrd  6187  lubfval  15262  glbfval  15275  oduglb  15423  odulub  15425  cnlnadjlem5  25622  cdj3lem3  25989  cdj3lem3b  25991  lshpkrlem1  33074  cdleme25cv  34321  cdlemk35  34875
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