Theorem ralunsn 3724

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 3257	. 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 3564	. . 3 |- ( B e. C -> ( A. x e. { B } ph <-> ps ) )
4	3	anbi2d 459	. 2 |- ( B e. C -> ( ( A. x e. A ph /\ A. x e. { B } ph ) <-> ( A. x e. A ph /\ ps ) ) )
5	1, 4	syl5bb 191	1 |- ( B e. C -> ( A. x e. ( A u. { B } ) ph <-> ( A. x e. A ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2416   u. cun 3069   {csn 3527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533

This theorem is referenced by:  2ralunsn  3725