Theorem riotaeqbidv 5627

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 5626	. 2 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
3		riotaeqbidv.1	. . 3 |- ( ph -> A = B )
4	3	riotaeqdv 5625	. 2 |- ( ph -> ( iota_ x e. A ch ) = ( iota_ x e. B ch ) )
5	2, 4	eqtrd 2121	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1290   iota_crio 5623

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-uni 3662  df-iota 4995  df-riota 5624

This theorem is referenced by:  acexmidlemab  5662