Theorem ralunsn 3589

Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
ralunsn.1	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralunsn	|- ( B e. C -> ( A. x e. ( A u. { B } ) ph <-> ( A. x e. A ph /\ ps ) ) )

Distinct variable groups:   x, B   ps, x

Allowed substitution hints:   ph( x)   A( x)   C( x)

Proof of Theorem ralunsn

Step	Hyp	Ref	Expression
1		ralunb 3153	. 2 |- ( A. x e. ( A u. { B } ) ph <-> ( A. x e. A ph /\ A. x e. { B } ph ) )
2		ralunsn.1	. . . 4 |- ( x = B -> ( ph <-> ps ) )
3	2	ralsng 3433	. . 3 |- ( B e. C -> ( A. x e. { B } ph <-> ps ) )
4	3	anbi2d 451	. 2 |- ( B e. C -> ( ( A. x e. A ph /\ A. x e. { B } ph ) <-> ( A. x e. A ph /\ ps ) ) )
5	1, 4	syl5bb 190	1 |- ( B e. C -> ( A. x e. ( A u. { B } ) ph <-> ( A. x e. A ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   u. cun 2971   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404

This theorem is referenced by:  2ralunsn  3590