Theorem riotaeqbidv 5973

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 5972	. 2 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
3		riotaeqbidv.1	. . 3 |- ( ph -> A = B )
4	3	riotaeqdv 5971	. 2 |- ( ph -> ( iota_ x e. A ch ) = ( iota_ x e. B ch ) )
5	2, 4	eqtrd 2264	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   iota_crio 5969

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-uni 3894  df-iota 5286  df-riota 5970

This theorem is referenced by:  acexmidlemab  6011  grpinvfvalg  13624  opprnegg  14095