Theorem riotaeqbidv 5957

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 5956	. 2 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
3		riotaeqbidv.1	. . 3 |- ( ph -> A = B )
4	3	riotaeqdv 5955	. 2 |- ( ph -> ( iota_ x e. A ch ) = ( iota_ x e. B ch ) )
5	2, 4	eqtrd 2262	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   iota_crio 5953

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3889  df-iota 5278  df-riota 5954

This theorem is referenced by:  acexmidlemab  5995  grpinvfvalg  13575  opprnegg  14046