Theorem ralunsn 3838

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 3354	. 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 3673	. . 3 |- ( B e. C -> ( A. x e. { B } ph <-> ps ) )
4	3	anbi2d 464	. 2 |- ( B e. C -> ( ( A. x e. A ph /\ A. x e. { B } ph ) <-> ( A. x e. A ph /\ ps ) ) )
5	1, 4	bitrid 192	1 |- ( B e. C -> ( A. x e. ( A u. { B } ) ph <-> ( A. x e. A ph /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   u. cun 3164   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639

This theorem is referenced by:  2ralunsn  3839  nnnninfeq2  7231