Theorem sb8eu 1017

Description: Variable substitution in uniqueness quantifier. (This theorem can also be proved without requiring that x and y be distinct, but the proof would be longer.)

Hypothesis

Ref	Expression
sb8eu.1	|- (ph -> A.yph)

Assertion

Ref	Expression
sb8eu	|- (E!xph <-> E!y[y / x]ph)

Distinct variable group(s):   x,y

Proof of Theorem sb8eu

Step	Hyp	Ref	Expression
1		sb8eu.1	. . . . . 6 |- (ph -> A.yph)
2		ax-17 925	. . . . . 6 |- (x = z -> A.y x = z)
3	1, 2	hbbi 705	. . . . 5 |- ((ph <-> x = z) -> A.y(ph <-> x = z))
4	3	sb8 918	. . . 4 |- (A.x(ph <-> x = z) <-> A.y[y / x](ph <-> x = z))
5		ax-17 925	. . . . . . 7 |- (y = z -> A.x y = z)
6		a8b 817	. . . . . . 7 |- (x = y -> (x = z <-> y = z))
7	5, 6	sbie 904	. . . . . 6 |- ([y / x]x = z <-> y = z)
8	7	sblbis 891	. . . . 5 |- ([y / x](ph <-> x = z) <-> ([y / x]ph <-> y = z))
9	8	bial 695	. . . 4 |- (A.y[y / x](ph <-> x = z) <-> A.y([y / x]ph <-> y = z))
10	4, 9	bitr 151	. . 3 |- (A.x(ph <-> x = z) <-> A.y([y / x]ph <-> y = z))
11	10	biex 733	. 2 |- (E.zA.x(ph <-> x = z) <-> E.zA.y([y / x]ph <-> y = z))
12		df-eu 1009	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
13		df-eu 1009	. 2 |- (E!y[y / x]ph <-> E.zA.y([y / x]ph <-> y = z))
14	11, 12, 13	3bitr4 158	1 |- (E!xph <-> E!y[y / x]ph)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  E.wex 678   = weq 797  [wsb 852  E!weu 1007

This theorem is referenced by:  cbveu 1018  eu1 1019

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-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009

metamath.org