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Theorem riotabiia 6249
Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 3095 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 2454 . 2  |-  _V  =  _V
2 riotabiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
32adantl 464 . . 3  |-  ( ( _V  =  _V  /\  x  e.  A )  ->  ( ph  <->  ps )
)
43riotabidva 6248 . 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 1398    e. wcel 1823   _Vcvv 3106   iota_crio 6231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-uni 4236  df-iota 5534  df-riota 6232
This theorem is referenced by:  riotaxfrd  6262  lubfval  15807  glbfval  15820  oduglb  15968  odulub  15970  cnlnadjlem5  27188  cdj3lem3  27555  cdj3lem3b  27557  lshpkrlem1  35232  cdleme25cv  36481  cdlemk35  37035
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