Theorem reuxfrd 3846

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. Use reuhypd 3848 to eliminate the second hypothesis.

Hypotheses

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

Assertion

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

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

Proof of Theorem reuxfrd

Step	Hyp	Ref	Expression
1		reuxfrd.2	. . . . . 6 |- ((ph /\ x e. B) -> E!y e. B x = A)
2		reurex 2440	. . . . . 6 |- (E!y e. B x = A -> E.y e. B x = A)
3	1, 2	syl 12	. . . . 5 |- ((ph /\ x e. B) -> E.y e. B x = A)
4	3	biantrurd 796	. . . 4 |- ((ph /\ x e. B) -> (ps <-> (E.y e. B x = A /\ ps)))
5		r19.41v 2236	. . . . 5 |- (E.y e. B (x = A /\ ps) <-> (E.y e. B x = A /\ ps))
6		reuxfrd.3	. . . . . . 7 |- (x = A -> (ps <-> ch))
7	6	pm5.32i 707	. . . . . 6 |- ((x = A /\ ps) <-> (x = A /\ ch))
8	7	rexbii 2128	. . . . 5 |- (E.y e. B (x = A /\ ps) <-> E.y e. B (x = A /\ ch))
9	5, 8	bitr3i 192	. . . 4 |- ((E.y e. B x = A /\ ps) <-> E.y e. B (x = A /\ ch))
10	4, 9	syl6bb 595	. . 3 |- ((ph /\ x e. B) -> (ps <-> E.y e. B (x = A /\ ch)))
11	10	reubidva 2259	. 2 |- (ph -> (E!x e. B ps <-> E!x e. B E.y e. B (x = A /\ ch)))
12		reuxfrd.1	. . 3 |- ((ph /\ y e. B) -> A e. B)
13		df-reu 2111	. . . . 5 |- (E!y e. B x = A <-> E!y(y e. B /\ x = A))
14		eumo 1807	. . . . 5 |- (E!y(y e. B /\ x = A) -> E*y(y e. B /\ x = A))
15	13, 14	sylbi 216	. . . 4 |- (E!y e. B x = A -> E*y(y e. B /\ x = A))
16	1, 15	syl 12	. . 3 |- ((ph /\ x e. B) -> E*y(y e. B /\ x = A))
17	12, 16	reuxfr2d 3844	. 2 |- (ph -> (E!x e. B E.y e. B (x = A /\ ch) <-> E!y e. B ch))
18	11, 17	bitrd 587	1 |- (ph -> (E!x e. B ps <-> E!y e. B ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  E*wmo 1772  E.wrex 2106  E!wreu 2107

This theorem is referenced by:  reuxfr 3847  riotaxfrd 5581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294

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