Theorem riotabiia 5485

Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2547 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 2040	. 2 |- _V = _V
2		riotabiia.1	. . . 4 |- ( x e. A -> ( ph <-> ps ) )
3	2	adantl 262	. . 3 |- ( ( _V = _V /\ x e. A ) -> ( ph <-> ps ) )
4	3	riotabidva 5484	. 2 |- ( _V = _V -> ( iota_ x e. A ph ) = ( iota_ x e. A ps ) )
5	1, 4	ax-mp 7	1 |- ( iota_ x e. A ph ) = ( iota_ x e. A ps )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   iota_crio 5467

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581  df-iota 4867  df-riota 5468

This theorem is referenced by:  caucvgsrlemfv  6875