Theorem reu8 1932

Description: Restricted uniqueness using implicit substitution.

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

Proof of Theorem reu8

Step	Hyp	Ref	Expression
1		rmo4.1	. . 3 |- (x = y -> (ph <-> ps))
2	1	cbvreuv 1798	. 2 |- (E!x e. A ph <-> E!y e. A ps)
3		reu3 1927	. 2 |- (E!y e. A ps <-> E.x e. A A.y e. A (ps <-> y = x))
4		eqcom 1474	. . . . . . . . . 10 |- (x = y <-> y = x)
5	4	imbi2i 185	. . . . . . . . 9 |- ((ps -> x = y) <-> (ps -> y = x))
6	5	ralbii 1664	. . . . . . . 8 |- (A.y e. A (ps -> x = y) <-> A.y e. A (ps -> y = x))
7	6	a1i 8	. . . . . . 7 |- (x e. A -> (A.y e. A (ps -> x = y) <-> A.y e. A (ps -> y = x)))
8		biimt 730	. . . . . . . 8 |- (x e. A -> (ph <-> (x e. A -> ph)))
9		df-ral 1646	. . . . . . . . 9 |- (A.y e. A (y = x -> ps) <-> A.y(y e. A -> (y = x -> ps)))
10		bi2.04 160	. . . . . . . . . 10 |- ((y e. A -> (y = x -> ps)) <-> (y = x -> (y e. A -> ps)))
11	10	albii 997	. . . . . . . . 9 |- (A.y(y e. A -> (y = x -> ps)) <-> A.y(y = x -> (y e. A -> ps)))
12		visset 1809	. . . . . . . . . 10 |- x e. V
13		eleq1 1531	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
14	13, 1	imbi12d 625	. . . . . . . . . . . 12 |- (x = y -> ((x e. A -> ph) <-> (y e. A -> ps)))
15	14	bicomd 520	. . . . . . . . . . 11 |- (x = y -> ((y e. A -> ps) <-> (x e. A -> ph)))
16	15	eqcoms 1475	. . . . . . . . . 10 |- (y = x -> ((y e. A -> ps) <-> (x e. A -> ph)))
17	12, 16	ceqsalv 1823	. . . . . . . . 9 |- (A.y(y = x -> (y e. A -> ps)) <-> (x e. A -> ph))
18	9, 11, 17	3bitrr 178	. . . . . . . 8 |- ((x e. A -> ph) <-> A.y e. A (y = x -> ps))
19	8, 18	syl6bb 535	. . . . . . 7 |- (x e. A -> (ph <-> A.y e. A (y = x -> ps)))
20	7, 19	anbi12d 627	. . . . . 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))))
21		ancom 435	. . . . . 6 |- ((ph /\ A.y e. A (ps -> x = y)) <-> (A.y e. A (ps -> x = y) /\ ph))
22	20, 21	syl5bb 531	. . . . 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))))
23		r19.26 1747	. . . . 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)))
24	22, 23	syl6rbbr 538	. . . 4 |- (x e. A -> (A.y e. A ((ps -> y = x) /\ (y = x -> ps)) <-> (ph /\ A.y e. A (ps -> x = y))))
25		bi 514	. . . . 5 |- ((ps <-> y = x) <-> ((ps -> y = x) /\ (y = x -> ps)))
26	25	ralbii 1664	. . . 4 |- (A.y e. A (ps <-> y = x) <-> A.y e. A ((ps -> y = x) /\ (y = x -> ps)))
27	24, 26	syl5bb 531	. . 3 |- (x e. A -> (A.y e. A (ps <-> y = x) <-> (ph /\ A.y e. A (ps -> x = y))))
28	27	rexbiia 1671	. 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	3bitr 177	1 |- (E!x e. A ph <-> E.x e. A (ph /\ A.y e. A (ps -> x = y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643  E!wreu 1644

This theorem is referenced by:  grpideu 8003  grpinveu 8014

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-reu 1648  df-v 1808

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