MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotabiia Unicode version

Theorem riotabiia 6322
Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2778 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 2283 . 2  |-  _V  =  _V
2 riotabiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
32adantl 452 . . 3  |-  ( ( _V  =  _V  /\  x  e.  A )  ->  ( ph  <->  ps )
)
43riotabidva 6321 . 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 1623    e. wcel 1684   _Vcvv 2788   iota_crio 6297
This theorem is referenced by:  riotaxfrd  6336  oduglb  14243  odulub  14245  cnlnadjlem5  22651  cdj3lem3  23018  cdj3lem3b  23020  lshpkrlem1  29300  cdleme25cv  30547  cdlemk35  31101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-un 3157  df-if 3566  df-uni 3828  df-iota 5219  df-riota 6304
  Copyright terms: Public domain W3C validator