Theorem ralunsn 3827

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 3344	. 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 3662	. . 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 1364   e. wcel 2167   A.wral 2475   u. cun 3155   {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628

This theorem is referenced by:  2ralunsn  3828  nnnninfeq2  7195