Theorem reu7 3032

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

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

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

Proof of Theorem reu7

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reu3 3027	. 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 2359	. . . . . . . 8 |- (x = y -> (x = z <-> y = z))
4		eqcom 2355	. . . . . . . 8 |- (y = z <-> z = y)
5	3, 4	syl6bb 252	. . . . . . 7 |- (x = y -> (x = z <-> z = y))
6	2, 5	imbi12d 311	. . . . . 6 |- (x = y -> ((ph -> x = z) <-> (ps -> z = y)))
7	6	cbvralv 2836	. . . . 5 |- (A.x e. A (ph -> x = z) <-> A.y e. A (ps -> z = y))
8	7	rexbii 2640	. . . 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 2359	. . . . . . 7 |- (z = x -> (z = y <-> x = y))
10	9	imbi2d 307	. . . . . 6 |- (z = x -> ((ps -> z = y) <-> (ps -> x = y)))
11	10	ralbidv 2635	. . . . 5 |- (z = x -> (A.y e. A (ps -> z = y) <-> A.y e. A (ps -> x = y)))
12	11	cbvrexv 2837	. . . 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	bitri 240	. . 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 675	. 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	bitri 240	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 4   <-> wb 176   /\ wa 358   = wceq 1642  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-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by: (None)