Theorem ralunsn 3880

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 3445	. 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 3766	. . 3 |- (B e. C -> (A.x e. {B}ph <-> ps))
4	3	anbi2d 684	. 2 |- (B e. C -> ((A.x e. A ph /\ A.x e. {B}ph) <-> (A.x e. A ph /\ ps)))
5	1, 4	syl5bb 248	1 |- (B e. C -> (A.x e. (A u. {B})ph <-> (A.x e. A ph /\ ps)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615   u. cun 3208  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742

This theorem is referenced by:  2ralunsn  3881