Theorem euxfr 1924

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 1393	. . . . . 6 |- (E!y x = A -> E.y x = A)
3	1, 2	ax-mp 7	. . . . 5 |- E.y x = A
4	3	biantrur 724	. . . 4 |- (ph <-> (E.y x = A /\ ph))
5		19.41v 1304	. . . 4 |- (E.y(x = A /\ ph) <-> (E.y x = A /\ ph))
6		euxfr.3	. . . . . 6 |- (x = A -> (ph <-> ps))
7	6	pm5.32i 644	. . . . 5 |- ((x = A /\ ph) <-> (x = A /\ ps))
8	7	exbii 1050	. . . 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 1386	. 2 |- (E!xph <-> E!xE.y(x = A /\ ps))
11		euxfr.1	. . 3 |- A e. V
12	1	eumoi 1411	. . 3 |- E*y x = A
13	11, 12	euxfr2 1923	. 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 955   e. wcel 957  E.wex 979  E!weu 1379  Vcvv 1808

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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