Theorem ralunsn 3690

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 3223	. 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 3530	. . 3 |- ( B e. C -> ( A. x e. { B } ph <-> ps ) )
4	3	anbi2d 457	. 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 1314   e. wcel 1463   A.wral 2390   u. cun 3035   {csn 3493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-sbc 2879  df-un 3041  df-sn 3499

This theorem is referenced by:  2ralunsn  3691