Theorem riotabidva 5754

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2677 analog.) (Contributed by NM, 17-Jan-2012.)

Hypothesis

Ref	Expression
riotabidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
riotabidva	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem riotabidva

Step	Hyp	Ref	Expression
1		riotabidva.1	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	pm5.32da 448	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
3	2	iotabidv 5117	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
4		df-riota 5738	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
5		df-riota 5738	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
6	3, 4, 5	3eqtr4g 2198	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   iotacio 5094   iota_crio 5737

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-uni 3745  df-iota 5096  df-riota 5738

This theorem is referenced by:  riotabiia  5755  divfnzn  9440