Theorem reu7 1935

Description: Restricted uniqueness using implicit substitution.

Hypothesis

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

Assertion

Ref	Expression
reu7	|- (E!x e. A ph <-> (E.x e. A ph /\ E.x e. A A.y e. A (ps -> x = y)))

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

Proof of Theorem reu7

Step	Hyp	Ref	Expression
1		reu6 1932	. 2 |- (E!x e. A ph <-> (E.x e. A ph /\ E.z e. A A.x e. A (ph -> x = z)))
2		rmo4.1	. . . . . . 7 |- (x = y -> (ph <-> ps))
3		eqeq1 1481	. . . . . . . 8 |- (x = y -> (x = z <-> y = z))
4		eqcom 1477	. . . . . . . 8 |- (y = z <-> z = y)
5	3, 4	syl6bb 536	. . . . . . 7 |- (x = y -> (x = z <-> z = y))
6	2, 5	imbi12d 626	. . . . . 6 |- (x = y -> ((ph -> x = z) <-> (ps -> z = y)))
7	6	cbvralv 1800	. . . . 5 |- (A.x e. A (ph -> x = z) <-> A.y e. A (ps -> z = y))
8	7	rexbii 1668	. . . 4 |- (E.z e. A A.x e. A (ph -> x = z) <-> E.z e. A A.y e. A (ps -> z = y))
9		eqeq1 1481	. . . . . . 7 |- (z = x -> (z = y <-> x = y))
10	9	imbi2d 612	. . . . . 6 |- (z = x -> ((ps -> z = y) <-> (ps -> x = y)))
11	10	ralbidv 1663	. . . . 5 |- (z = x -> (A.y e. A (ps -> z = y) <-> A.y e. A (ps -> x = y)))
12	11	cbvrexv 1801	. . . 4 |- (E.z e. A A.y e. A (ps -> z = y) <-> E.x e. A A.y e. A (ps -> x = y))
13	8, 12	bitr 173	. . 3 |- (E.z e. A A.x e. A (ph -> x = z) <-> E.x e. A A.y e. A (ps -> x = y))
14	13	anbi2i 480	. 2 |- ((E.x e. A ph /\ E.z e. A A.x e. A (ph -> x = z)) <-> (E.x e. A ph /\ E.x e. A A.y e. A (ps -> x = y)))
15	1, 14	bitr 173	1 |- (E!x e. A ph <-> (E.x e. A ph /\ E.x e. A A.y e. A (ps -> x = y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  A.wral 1645  E.wrex 1646  E!wreu 1647

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-reu 1651

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