Theorem reuxfr2 3845

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

Hypotheses

Ref	Expression
reuxfr2.1	|- (y e. B -> A e. B)
reuxfr2.2	|- (x e. B -> E*y(y e. B /\ x = A))

Assertion

Ref	Expression
reuxfr2	|- (E!x e. B E.y e. B (x = A /\ ph) <-> E!y e. B ph)

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

Proof of Theorem reuxfr2

Step	Hyp	Ref	Expression
1		2reuswap 2449	. . . 4 |- (A.x e. B E*y(y e. B /\ (x = A /\ ph)) -> (E!x e. B E.y e. B (x = A /\ ph) -> E!y e. B E.x e. B (x = A /\ ph)))
2		reuxfr2.2	. . . . . 6 |- (x e. B -> E*y(y e. B /\ x = A))
3		moan 1819	. . . . . 6 |- (E*y(y e. B /\ x = A) -> E*y(ph /\ (y e. B /\ x = A)))
4	2, 3	syl 12	. . . . 5 |- (x e. B -> E*y(ph /\ (y e. B /\ x = A)))
5		ancom 482	. . . . . . 7 |- ((ph /\ (y e. B /\ x = A)) <-> ((y e. B /\ x = A) /\ ph))
6		anass 487	. . . . . . 7 |- (((y e. B /\ x = A) /\ ph) <-> (y e. B /\ (x = A /\ ph)))
7	5, 6	bitri 190	. . . . . 6 |- ((ph /\ (y e. B /\ x = A)) <-> (y e. B /\ (x = A /\ ph)))
8	7	mobii 1801	. . . . 5 |- (E*y(ph /\ (y e. B /\ x = A)) <-> E*y(y e. B /\ (x = A /\ ph)))
9	4, 8	sylib 215	. . . 4 |- (x e. B -> E*y(y e. B /\ (x = A /\ ph)))
10	1, 9	mprg 2162	. . 3 |- (E!x e. B E.y e. B (x = A /\ ph) -> E!y e. B E.x e. B (x = A /\ ph))
11		2reuswap 2449	. . . 4 |- (A.y e. B E*x(x e. B /\ (x = A /\ ph)) -> (E!y e. B E.x e. B (x = A /\ ph) -> E!x e. B E.y e. B (x = A /\ ph)))
12		moeq 2431	. . . . . . 7 |- E*x x = A
13	12	moani 1820	. . . . . 6 |- E*x((x e. B /\ ph) /\ x = A)
14		ancom 482	. . . . . . . 8 |- (((x e. B /\ ph) /\ x = A) <-> (x = A /\ (x e. B /\ ph)))
15		an12 542	. . . . . . . 8 |- ((x = A /\ (x e. B /\ ph)) <-> (x e. B /\ (x = A /\ ph)))
16	14, 15	bitri 190	. . . . . . 7 |- (((x e. B /\ ph) /\ x = A) <-> (x e. B /\ (x = A /\ ph)))
17	16	mobii 1801	. . . . . 6 |- (E*x((x e. B /\ ph) /\ x = A) <-> E*x(x e. B /\ (x = A /\ ph)))
18	13, 17	mpbi 206	. . . . 5 |- E*x(x e. B /\ (x = A /\ ph))
19	18	a1i 8	. . . 4 |- (y e. B -> E*x(x e. B /\ (x = A /\ ph)))
20	11, 19	mprg 2162	. . 3 |- (E!y e. B E.x e. B (x = A /\ ph) -> E!x e. B E.y e. B (x = A /\ ph))
21	10, 20	impbii 174	. 2 |- (E!x e. B E.y e. B (x = A /\ ph) <-> E!y e. B E.x e. B (x = A /\ ph))
22		reuxfr2.1	. . . 4 |- (y e. B -> A e. B)
23		biidd 188	. . . . 5 |- (x = A -> (ph <-> ph))
24	23	ceqsrexv 2394	. . . 4 |- (A e. B -> (E.x e. B (x = A /\ ph) <-> ph))
25	22, 24	syl 12	. . 3 |- (y e. B -> (E.x e. B (x = A /\ ph) <-> ph))
26	25	reubiia 2261	. 2 |- (E!y e. B E.x e. B (x = A /\ ph) <-> E!y e. B ph)
27	21, 26	bitri 190	1 |- (E!x e. B E.y e. B (x = A /\ ph) <-> E!y e. B ph)

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

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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