Theorem riotabiia 5838

Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2720 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 2175	. 2 |- _V = _V
2		riotabiia.1	. . . 4 |- ( x e. A -> ( ph <-> ps ) )
3	2	adantl 277	. . 3 |- ( ( _V = _V /\ x e. A ) -> ( ph <-> ps ) )
4	3	riotabidva 5837	. 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 105   = wceq 1353   e. wcel 2146   _Vcvv 2735   iota_crio 5820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-uni 3806  df-iota 5170  df-riota 5821

This theorem is referenced by:  caucvgsrlemfv  7765  dfgcd3  11976