Theorem bieud 1012

Description: Formula-building rule for uniqueness quantifier (deduction rule).

Hypotheses

Ref	Expression
bieud.1	|- (ph -> A.xph)
bieud.2	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
bieud	|- (ph -> (E!xps <-> E!xch))

Proof of Theorem bieud

Step	Hyp	Ref	Expression
1		bieud.1	. . . 4 |- (ph -> A.xph)
2		bieud.2	. . . . 5 |- (ph -> (ps <-> ch))
3	2	bibi1d 471	. . . 4 |- (ph -> ((ps <-> x = y) <-> (ch <-> x = y)))
4	1, 3	biald 782	. . 3 |- (ph -> (A.x(ps <-> x = y) <-> A.x(ch <-> x = y)))
5	4	biexdv 936	. 2 |- (ph -> (E.yA.x(ps <-> x = y) <-> E.yA.x(ch <-> x = y)))
6		df-eu 1009	. 2 |- (E!xps <-> E.yA.x(ps <-> x = y))
7		df-eu 1009	. 2 |- (E!xch <-> E.yA.x(ch <-> x = y))
8	5, 6, 7	3bitr4g 428	1 |- (ph -> (E!xps <-> E!xch))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  E.wex 678   = weq 797  E!weu 1007

This theorem is referenced by:  bieudv 1013  bieu 1014  bimod 1030  reueqf 1323

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009

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