Theorem reuxfr 2900

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. Use reuhyp 2901 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 1925	. . . . . 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 726	. . . 4 |- (x e. B -> (ph <-> (E.y e. B x = A /\ ph)))
5		r19.41v 1761	. . . . 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 644	. . . . . 6 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	rexbii 1666	. . . . 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 535	. . 3 |- (x e. B -> (ph <-> E.y e. B (x = A /\ ps)))
11	10	reubiia 1779	. 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 1649	. . . . 5 |- (E!y e. B x = A <-> E!y(y e. B /\ x = A))
14		eumo 1410	. . . . 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 2899	. 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 955   e. wcel 957  E!weu 1379  E*wmo 1380  E.wrex 1644  E!wreu 1645

This theorem is referenced by:  reuunixfr 2902  zmax 6178  rebtwnz 6180

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-rex 1648  df-reu 1649  df-v 1809

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