Theorem riotabiia 6628

Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 3185 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

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- _V = _V
2		riotabiia.1	. . . 4 |- ( x e. A -> ( ph <-> ps ) )
3	2	adantl 482	. . 3 |- ( ( _V = _V /\ x e. A ) -> ( ph <-> ps ) )
4	3	riotabidva 6627	. 2 |- ( _V = _V -> ( iota_ x e. A ph ) = ( iota_ x e. A ps ) )
5	1, 4	ax-mp 5	1 |- ( iota_ x e. A ph ) = ( iota_ x e. A ps )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   iota_crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-uni 4437  df-iota 5851  df-riota 6611

This theorem is referenced by:  riotaxfrd  6642  lubfval  16978  glbfval  16991  oduglb  17139  odulub  17141  cnlnadjlem5  28930  cdj3lem3  29297  cdj3lem3b  29299  lshpkrlem1  34397  cdleme25cv  35646  cdlemk35  36200