Theorem riotaeqbidv 5471

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)

Hypotheses

Ref	Expression
riotaeqbidv.1	|- ( ph -> A = B )
riotaeqbidv.2	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Distinct variable group:   ph, x

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

Proof of Theorem riotaeqbidv

Step	Hyp	Ref	Expression
1		riotaeqbidv.2	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	riotabidv 5470	. 2 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
3		riotaeqbidv.1	. . 3 |- ( ph -> A = B )
4	3	riotaeqdv 5469	. 2 |- ( ph -> ( iota_ x e. A ch ) = ( iota_ x e. B ch ) )
5	2, 4	eqtrd 2072	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   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:  acexmidlemab  5506