Theorem reu8 2448

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 2282	. 2 |- (E!x e. A ph <-> E!y e. A ps)
3		reu6 2443	. 2 |- (E!y e. A ps <-> E.x e. A A.y e. A (ps <-> y = x))
4		eqcom 1886	. . . . . . . . . 10 |- (x = y <-> y = x)
5	4	imbi2i 202	. . . . . . . . 9 |- ((ps -> x = y) <-> (ps -> y = x))
6	5	ralbii 2127	. . . . . . . 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 803	. . . . . . . 8 |- (x e. A -> (ph <-> (x e. A -> ph)))
9		df-ral 2109	. . . . . . . . 9 |- (A.y e. A (y = x -> ps) <-> A.y(y e. A -> (y = x -> ps)))
10		bi2.04 177	. . . . . . . . . 10 |- ((y e. A -> (y = x -> ps)) <-> (y = x -> (y e. A -> ps)))
11	10	albii 1346	. . . . . . . . 9 |- (A.y(y e. A -> (y = x -> ps)) <-> A.y(y = x -> (y e. A -> ps)))
12		visset 2295	. . . . . . . . . 10 |- x e. _V
13		eleq1 1957	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
14	13, 1	imbi12d 688	. . . . . . . . . . . 12 |- (x = y -> ((x e. A -> ph) <-> (y e. A -> ps)))
15	14	bicomd 580	. . . . . . . . . . 11 |- (x = y -> ((y e. A -> ps) <-> (x e. A -> ph)))
16	15	eqcoms 1887	. . . . . . . . . 10 |- (y = x -> ((y e. A -> ps) <-> (x e. A -> ph)))
17	12, 16	ceqsalv 2317	. . . . . . . . 9 |- (A.y(y = x -> (y e. A -> ps)) <-> (x e. A -> ph))
18	9, 11, 17	3bitrri 195	. . . . . . . 8 |- ((x e. A -> ph) <-> A.y e. A (y = x -> ps))
19	8, 18	syl6bb 595	. . . . . . 7 |- (x e. A -> (ph <-> A.y e. A (y = x -> ps)))
20	7, 19	anbi12d 690	. . . . . 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 482	. . . . . 6 |- ((ph /\ A.y e. A (ps -> x = y)) <-> (A.y e. A (ps -> x = y) /\ ph))
22	20, 21	syl5bb 591	. . . . 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 2219	. . . . 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 598	. . . 4 |- (x e. A -> (A.y e. A ((ps -> y = x) /\ (y = x -> ps)) <-> (ph /\ A.y e. A (ps -> x = y))))
25		dfbi2 572	. . . . 5 |- ((ps <-> y = x) <-> ((ps -> y = x) /\ (y = x -> ps)))
26	25	ralbii 2127	. . . 4 |- (A.y e. A (ps <-> y = x) <-> A.y e. A ((ps -> y = x) /\ (y = x -> ps)))
27	24, 26	syl5bb 591	. . 3 |- (x e. A -> (A.y e. A (ps <-> y = x) <-> (ph /\ A.y e. A (ps -> x = y))))
28	27	rexbiia 2134	. 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 194	1 |- (E!x e. A ph <-> E.x e. A (ph /\ A.y e. A (ps -> x = y)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107

This theorem is referenced by:  grpideu 9333  grpinveu 9348  exidu1 10373  grpideuNEW 17114  grpinveuNEW 17123

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294

Copyright terms: Public domain