Theorem reuxfr 1580

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A.

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 group(s):   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 1337	. . . . . 6 |- (E!y e. B x = A -> E.y e. B x = A)
3	1, 2	syl 12	. . . . 5 |- (x e. B -> E.y e. B x = A)
4	3	biantrurd 546	. . . 4 |- (x e. B -> (ph <-> (E.y e. B x = A /\ ph)))
5		r19.41v 1302	. . . . 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 489	. . . . . 6 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	birex 1224	. . . . 5 |- (E.y e. B (x = A /\ ph) <-> E.y e. B (x = A /\ ps))
9	5, 8	bitr3 153	. . . 4 |- ((E.y e. B x = A /\ ph) <-> E.y e. B (x = A /\ ps))
10	4, 9	syl6bb 414	. . 3 |- (x e. B -> (ph <-> E.y e. B (x = A /\ ps)))
11	10	bireua 1319	. 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 1207	. . . . 5 |- (E!y e. B x = A <-> E!y(y e. B /\ x = A))
14		eumo 1037	. . . . 5 |- (E!y(y e. B /\ x = A) -> E*y(y e. B /\ x = A))
15	13, 14	sylbi 174	. . . 4 |- (E!y e. B x = A -> E*y(y e. B /\ x = A))
16	1, 15	syl 12	. . 3 |- (x e. B -> E*y(y e. B /\ x = A))
17	12, 16	reuxfr2 1579	. 2 |- (E!x e. B E.y e. B (x = A /\ ps) <-> E!y e. B ps)
18	11, 17	bitr 151	1 |- (E!x e. B ph <-> E!y e. B ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E!weu 1007  E*wmo 1008   = wceq 1091   e. wcel 1092  E.wrex 1202  E!wreu 1203

This theorem is referenced by:  zmax 4618  rebtwnz 4620

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-v 1349

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