Theorem euxfr 2438

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

Hypotheses

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

Assertion

Ref	Expression
euxfr	|- (E!xph <-> E!yps)

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

Proof of Theorem euxfr

Step	Hyp	Ref	Expression
1		euxfr.2	. . . . . 6 |- E!y x = A
2		euex 1788	. . . . . 6 |- (E!y x = A -> E.y x = A)
3	1, 2	ax-mp 7	. . . . 5 |- E.y x = A
4	3	biantrur 794	. . . 4 |- (ph <-> (E.y x = A /\ ph))
5		19.41v 1685	. . . 4 |- (E.y(x = A /\ ph) <-> (E.y x = A /\ ph))
6		euxfr.3	. . . . . 6 |- (x = A -> (ph <-> ps))
7	6	pm5.32i 707	. . . . 5 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	exbii 1398	. . . 4 |- (E.y(x = A /\ ph) <-> E.y(x = A /\ ps))
9	4, 5, 8	3bitr2i 196	. . 3 |- (ph <-> E.y(x = A /\ ps))
10	9	eubii 1780	. 2 |- (E!xph <-> E!xE.y(x = A /\ ps))
11		euxfr.1	. . 3 |- A e. _V
12	1	eumoi 1808	. . 3 |- E*y x = A
13	11, 12	euxfr2 2437	. 2 |- (E!xE.y(x = A /\ ps) <-> E!yps)
14	10, 13	bitri 190	1 |- (E!xph <-> E!yps)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  _Vcvv 2292

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-v 2294

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