Theorem 2ralunsn 3881

Description: Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Hypotheses

Ref	Expression
2ralunsn.1	|- (x = B -> (ph <-> ch))
2ralunsn.2	|- (y = B -> (ph <-> ps))
2ralunsn.3	|- (x = B -> (ps <-> th))

Assertion

Ref	Expression
2ralunsn	|- (B e. C -> (A.x e. (A u. {B})A.y e. (A u. {B})ph <-> ((A.x e. A A.y e. A ph /\ A.x e. A ps) /\ (A.y e. A ch /\ th))))

Distinct variable groups:   x,A   x,B,y   x,C   ch,x   ps,y   th,x

Allowed substitution hints:   ph(x,y)   ps(x)   ch(y)   th(y)   A(y)   C(y)

Proof of Theorem 2ralunsn

Step	Hyp	Ref	Expression
1		2ralunsn.2	. . . 4 |- (y = B -> (ph <-> ps))
2	1	ralunsn 3880	. . 3 |- (B e. C -> (A.y e. (A u. {B})ph <-> (A.y e. A ph /\ ps)))
3	2	ralbidv 2635	. 2 |- (B e. C -> (A.x e. (A u. {B})A.y e. (A u. {B})ph <-> A.x e. (A u. {B})(A.y e. A ph /\ ps)))
4		2ralunsn.1	. . . . . 6 |- (x = B -> (ph <-> ch))
5	4	ralbidv 2635	. . . . 5 |- (x = B -> (A.y e. A ph <-> A.y e. A ch))
6		2ralunsn.3	. . . . 5 |- (x = B -> (ps <-> th))
7	5, 6	anbi12d 691	. . . 4 |- (x = B -> ((A.y e. A ph /\ ps) <-> (A.y e. A ch /\ th)))
8	7	ralunsn 3880	. . 3 |- (B e. C -> (A.x e. (A u. {B})(A.y e. A ph /\ ps) <-> (A.x e. A (A.y e. A ph /\ ps) /\ (A.y e. A ch /\ th))))
9		r19.26 2747	. . . 4 |- (A.x e. A (A.y e. A ph /\ ps) <-> (A.x e. A A.y e. A ph /\ A.x e. A ps))
10	9	anbi1i 676	. . 3 |- ((A.x e. A (A.y e. A ph /\ ps) /\ (A.y e. A ch /\ th)) <-> ((A.x e. A A.y e. A ph /\ A.x e. A ps) /\ (A.y e. A ch /\ th)))
11	8, 10	syl6bb 252	. 2 |- (B e. C -> (A.x e. (A u. {B})(A.y e. A ph /\ ps) <-> ((A.x e. A A.y e. A ph /\ A.x e. A ps) /\ (A.y e. A ch /\ th))))
12	3, 11	bitrd 244	1 |- (B e. C -> (A.x e. (A u. {B})A.y e. (A u. {B})ph <-> ((A.x e. A A.y e. A ph /\ A.x e. A ps) /\ (A.y e. A ch /\ th))))

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: (None)