Theorem reu8 3033

Description: Restricted unique existence using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Hypothesis

Ref	Expression
rmo4.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
reu8	|- (E!x e. A ph <-> E.x e. A (ph /\ A.y e. A (ps -> x = y)))

Distinct variable groups:   x,y,A   ph,y   ps,x

Allowed substitution hints:   ph(x)   ps(y)

Proof of Theorem reu8

Step	Hyp	Ref	Expression
1		rmo4.1	. . 3 |- (x = y -> (ph <-> ps))
2	1	cbvreuv 2838	. 2 |- (E!x e. A ph <-> E!y e. A ps)
3		reu6 3026	. 2 |- (E!y e. A ps <-> E.x e. A A.y e. A (ps <-> y = x))
4		dfbi2 609	. . . . 5 |- ((ps <-> y = x) <-> ((ps -> y = x) /\ (y = x -> ps)))
5	4	ralbii 2639	. . . 4 |- (A.y e. A (ps <-> y = x) <-> A.y e. A ((ps -> y = x) /\ (y = x -> ps)))
6		ancom 437	. . . . . 6 |- ((ph /\ A.y e. A (ps -> x = y)) <-> (A.y e. A (ps -> x = y) /\ ph))
7		equcom 1680	. . . . . . . . . 10 |- (x = y <-> y = x)
8	7	imbi2i 303	. . . . . . . . 9 |- ((ps -> x = y) <-> (ps -> y = x))
9	8	ralbii 2639	. . . . . . . 8 |- (A.y e. A (ps -> x = y) <-> A.y e. A (ps -> y = x))
10	9	a1i 10	. . . . . . 7 |- (x e. A -> (A.y e. A (ps -> x = y) <-> A.y e. A (ps -> y = x)))
11		biimt 325	. . . . . . . 8 |- (x e. A -> (ph <-> (x e. A -> ph)))
12		df-ral 2620	. . . . . . . . 9 |- (A.y e. A (y = x -> ps) <-> A.y(y e. A -> (y = x -> ps)))
13		bi2.04 350	. . . . . . . . . 10 |- ((y e. A -> (y = x -> ps)) <-> (y = x -> (y e. A -> ps)))
14	13	albii 1566	. . . . . . . . 9 |- (A.y(y e. A -> (y = x -> ps)) <-> A.y(y = x -> (y e. A -> ps)))
15		vex 2863	. . . . . . . . . 10 |- x e. _V
16		eleq1 2413	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
17	16, 1	imbi12d 311	. . . . . . . . . . . 12 |- (x = y -> ((x e. A -> ph) <-> (y e. A -> ps)))
18	17	bicomd 192	. . . . . . . . . . 11 |- (x = y -> ((y e. A -> ps) <-> (x e. A -> ph)))
19	18	equcoms 1681	. . . . . . . . . 10 |- (y = x -> ((y e. A -> ps) <-> (x e. A -> ph)))
20	15, 19	ceqsalv 2886	. . . . . . . . 9 |- (A.y(y = x -> (y e. A -> ps)) <-> (x e. A -> ph))
21	12, 14, 20	3bitrri 263	. . . . . . . 8 |- ((x e. A -> ph) <-> A.y e. A (y = x -> ps))
22	11, 21	syl6bb 252	. . . . . . 7 |- (x e. A -> (ph <-> A.y e. A (y = x -> ps)))
23	10, 22	anbi12d 691	. . . . . 6 |- (x e. A -> ((A.y e. A (ps -> x = y) /\ ph) <-> (A.y e. A (ps -> y = x) /\ A.y e. A (y = x -> ps))))
24	6, 23	syl5bb 248	. . . . 5 |- (x e. A -> ((ph /\ A.y e. A (ps -> x = y)) <-> (A.y e. A (ps -> y = x) /\ A.y e. A (y = x -> ps))))
25		r19.26 2747	. . . . 5 |- (A.y e. A ((ps -> y = x) /\ (y = x -> ps)) <-> (A.y e. A (ps -> y = x) /\ A.y e. A (y = x -> ps)))
26	24, 25	syl6rbbr 255	. . . 4 |- (x e. A -> (A.y e. A ((ps -> y = x) /\ (y = x -> ps)) <-> (ph /\ A.y e. A (ps -> x = y))))
27	5, 26	syl5bb 248	. . 3 |- (x e. A -> (A.y e. A (ps <-> y = x) <-> (ph /\ A.y e. A (ps -> x = y))))
28	27	rexbiia 2648	. 2 |- (E.x e. A A.y e. A (ps <-> y = x) <-> E.x e. A (ph /\ A.y e. A (ps -> x = y)))
29	2, 3, 28	3bitri 262	1 |- (E!x e. A ph <-> E.x e. A (ph /\ A.y e. A (ps -> x = y)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  A.wral 2615  E.wrex 2616  E!wreu 2617

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622  df-v 2862

This theorem is referenced by: (None)