Theorem riotaeqbidv 6034

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 6033	. 2 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
3		riotaeqbidv.1	. . 3 |- ( ph -> A = B )
4	3	riotaeqdv 6032	. 2 |- ( ph -> ( iota_ x e. A ch ) = ( iota_ x e. B ch ) )
5	2, 4	eqtrd 2271	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   iota_crio 6030

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3934  df-iota 5335  df-riota 6031

This theorem is referenced by:  acexmidlemab  6072  grpinvfvalg  13827  opprnegg  14365