Theorem sb8eu 1388

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:   x,y

Proof of Theorem sb8eu

Step	Hyp	Ref	Expression
1		sb8eu.1	. . . . . 6 |- (ph -> A.yph)
2		ax-17 969	. . . . . 6 |- (x = z -> A.y x = z)
3	1, 2	hbbi 1008	. . . . 5 |- ((ph <-> x = z) -> A.y(ph <-> x = z))
4	3	sb8 1259	. . . 4 |- (A.x(ph <-> x = z) <-> A.y[y / x](ph <-> x = z))
5		ax-17 969	. . . . . . 7 |- (y = z -> A.x y = z)
6		equequ1 1132	. . . . . . 7 |- (x = y -> (x = z <-> y = z))
7	5, 6	sbie 1194	. . . . . 6 |- ([y / x]x = z <-> y = z)
8	7	sblbis 1238	. . . . 5 |- ([y / x](ph <-> x = z) <-> ([y / x]ph <-> y = z))
9	8	albii 997	. . . 4 |- (A.y[y / x](ph <-> x = z) <-> A.y([y / x]ph <-> y = z))
10	4, 9	bitr 173	. . 3 |- (A.x(ph <-> x = z) <-> A.y([y / x]ph <-> y = z))
11	10	exbii 1049	. 2 |- (E.zA.x(ph <-> x = z) <-> E.zA.y([y / x]ph <-> y = z))
12		df-eu 1380	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
13		df-eu 1380	. 2 |- (E!y[y / x]ph <-> E.zA.y([y / x]ph <-> y = z))
14	11, 12, 13	3bitr4 183	1 |- (E!xph <-> E!y[y / x]ph)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952  E.wex 978  [wsbc 1168  E!weu 1378

This theorem is referenced by:  cbveu 1389  eu1 1390

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380

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