Theorem riotabiia 5815

Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2711 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 2165	. 2 |- _V = _V
2		riotabiia.1	. . . 4 |- ( x e. A -> ( ph <-> ps ) )
3	2	adantl 275	. . 3 |- ( ( _V = _V /\ x e. A ) -> ( ph <-> ps ) )
4	3	riotabidva 5814	. 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 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   iota_crio 5797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790  df-iota 5153  df-riota 5798

This theorem is referenced by:  caucvgsrlemfv  7732  dfgcd3  11943