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Theorem riotabiia 6338
Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2791 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 2296 . 2  |-  _V  =  _V
2 riotabiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
32adantl 452 . . 3  |-  ( ( _V  =  _V  /\  x  e.  A )  ->  ( ph  <->  ps )
)
43riotabidva 6337 . 2  |-  ( _V  =  _V  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  A ps )
)
51, 4ax-mp 8 1  |-  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  A ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   iota_crio 6313
This theorem is referenced by:  riotaxfrd  6352  oduglb  14259  odulub  14261  cnlnadjlem5  22667  cdj3lem3  23034  cdj3lem3b  23036  lshpkrlem1  29922  cdleme25cv  31169  cdlemk35  31723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-uni 3844  df-iota 5235  df-riota 6320
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