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Theorem riotabiia 6567
Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2946 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 2436 . 2  |-  _V  =  _V
2 riotabiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
32adantl 453 . . 3  |-  ( ( _V  =  _V  /\  x  e.  A )  ->  ( ph  <->  ps )
)
43riotabidva 6566 . 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 177    = wceq 1652    e. wcel 1725   _Vcvv 2956   iota_crio 6542
This theorem is referenced by:  riotaxfrd  6581  oduglb  14566  odulub  14568  cnlnadjlem5  23574  cdj3lem3  23941  cdj3lem3b  23943  lshpkrlem1  29908  cdleme25cv  31155  cdlemk35  31709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-un 3325  df-if 3740  df-uni 4016  df-iota 5418  df-riota 6549
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