Theorem reu8 1939

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 1805	. 2 |- (E!x e. A ph <-> E!y e. A ps)
3		reu3 1934	. 2 |- (E!y e. A ps <-> E.x e. A A.y e. A (ps <-> y = x))
4		eqcom 1480	. . . . . . . . . 10 |- (x = y <-> y = x)
5	4	imbi2i 185	. . . . . . . . 9 |- ((ps -> x = y) <-> (ps -> y = x))
6	5	ralbii 1670	. . . . . . . 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 733	. . . . . . . 8 |- (x e. A -> (ph <-> (x e. A -> ph)))
9		df-ral 1652	. . . . . . . . 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 1001	. . . . . . . . 9 |- (A.y(y e. A -> (y = x -> ps)) <-> A.y(y = x -> (y e. A -> ps)))
12		visset 1816	. . . . . . . . . 10 |- x e. V
13		eleq1 1537	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
14	13, 1	imbi12d 628	. . . . . . . . . . . 12 |- (x = y -> ((x e. A -> ph) <-> (y e. A -> ps)))
15	14	bicomd 523	. . . . . . . . . . 11 |- (x = y -> ((y e. A -> ps) <-> (x e. A -> ph)))
16	15	eqcoms 1481	. . . . . . . . . 10 |- (y = x -> ((y e. A -> ps) <-> (x e. A -> ph)))
17	12, 16	ceqsalv 1830	. . . . . . . . 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 538	. . . . . . 7 |- (x e. A -> (ph <-> A.y e. A (y = x -> ps)))
20	7, 19	anbi12d 630	. . . . . 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 437	. . . . . 6 |- ((ph /\ A.y e. A (ps -> x = y)) <-> (A.y e. A (ps -> x = y) /\ ph))
22	20, 21	syl5bb 534	. . . . 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 1753	. . . . 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 541	. . . 4 |- (x e. A -> (A.y e. A ((ps -> y = x) /\ (y = x -> ps)) <-> (ph /\ A.y e. A (ps -> x = y))))
25		dfbi2 516	. . . . 5 |- ((ps <-> y = x) <-> ((ps -> y = x) /\ (y = x -> ps)))
26	25	ralbii 1670	. . . 4 |- (A.y e. A (ps <-> y = x) <-> A.y e. A ((ps -> y = x) /\ (y = x -> ps)))
27	24, 26	syl5bb 534	. . 3 |- (x e. A -> (A.y e. A (ps <-> y = x) <-> (ph /\ A.y e. A (ps -> x = y))))
28	27	rexbiia 1677	. 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 956   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649  E!wreu 1650

This theorem is referenced by:  grpideu 8050  grpinveu 8060

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-reu 1654  df-v 1815

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