Theorem euxfr 1927

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 1394	. . . . . 6 |- (E!y x = A -> E.y x = A)
3	1, 2	ax-mp 7	. . . . 5 |- E.y x = A
4	3	biantrur 725	. . . 4 |- (ph <-> (E.y x = A /\ ph))
5		19.41v 1305	. . . 4 |- (E.y(x = A /\ ph) <-> (E.y x = A /\ ph))
6		euxfr.3	. . . . . 6 |- (x = A -> (ph <-> ps))
7	6	pm5.32i 645	. . . . 5 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	exbii 1051	. . . 4 |- (E.y(x = A /\ ph) <-> E.y(x = A /\ ps))
9	4, 5, 8	3bitr2 179	. . 3 |- (ph <-> E.y(x = A /\ ps))
10	9	eubii 1387	. 2 |- (E!xph <-> E!xE.y(x = A /\ ps))
11		euxfr.1	. . 3 |- A e. V
12	1	eumoi 1412	. . 3 |- E*y x = A
13	11, 12	euxfr2 1926	. 2 |- (E!xE.y(x = A /\ ps) <-> E!yps)
14	10, 13	bitr 173	1 |- (E!xph <-> E!yps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380  Vcvv 1811

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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