Theorem riotabiia 5826

Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2715 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 2170	. 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 5825	. 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 1348   e. wcel 2141   _Vcvv 2730   iota_crio 5808

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-uni 3797  df-iota 5160  df-riota 5809

This theorem is referenced by:  caucvgsrlemfv  7753  dfgcd3  11965