Theorem reuxfr 2894

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. Use reuhyp 2895 to eliminate the second hypothesis.

Hypotheses

Ref	Expression
reuxfr.1	|- (y e. B -> A e. B)
reuxfr.2	|- (x e. B -> E!y e. B x = A)
reuxfr.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
reuxfr	|- (E!x e. B ph <-> E!y e. B ps)

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

Proof of Theorem reuxfr

Step	Hyp	Ref	Expression
1		reuxfr.2	. . . . . 6 |- (x e. B -> E!y e. B x = A)
2		reurex 1918	. . . . . 6 |- (E!y e. B x = A -> E.y e. B x = A)
3	1, 2	syl 10	. . . . 5 |- (x e. B -> E.y e. B x = A)
4	3	biantrurd 725	. . . 4 |- (x e. B -> (ph <-> (E.y e. B x = A /\ ph)))
5		r19.41v 1755	. . . . 5 |- (E.y e. B (x = A /\ ph) <-> (E.y e. B x = A /\ ph))
6		reuxfr.3	. . . . . . 7 |- (x = A -> (ph <-> ps))
7	6	pm5.32i 643	. . . . . 6 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	rexbii 1660	. . . . 5 |- (E.y e. B (x = A /\ ph) <-> E.y e. B (x = A /\ ps))
9	5, 8	bitr3 175	. . . 4 |- ((E.y e. B x = A /\ ph) <-> E.y e. B (x = A /\ ps))
10	4, 9	syl6bb 534	. . 3 |- (x e. B -> (ph <-> E.y e. B (x = A /\ ps)))
11	10	reubiia 1773	. 2 |- (E!x e. B ph <-> E!x e. B E.y e. B (x = A /\ ps))
12		reuxfr.1	. . 3 |- (y e. B -> A e. B)
13		df-reu 1643	. . . . 5 |- (E!y e. B x = A <-> E!y(y e. B /\ x = A))
14		eumo 1404	. . . . 5 |- (E!y(y e. B /\ x = A) -> E*y(y e. B /\ x = A))
15	13, 14	sylbi 199	. . . 4 |- (E!y e. B x = A -> E*y(y e. B /\ x = A))
16	1, 15	syl 10	. . 3 |- (x e. B -> E*y(y e. B /\ x = A))
17	12, 16	reuxfr2 2893	. 2 |- (E!x e. B E.y e. B (x = A /\ ps) <-> E!y e. B ps)
18	11, 17	bitr 173	1 |- (E!x e. B ph <-> E!y e. B ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E!weu 1373  E*wmo 1374  E.wrex 1638  E!wreu 1639

This theorem is referenced by:  reuunixfr 2896  zmax 6168  rebtwnz 6170

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-reu 1643  df-v 1803

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