Theorem sb8eu 2222

Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypothesis

Ref	Expression
sb8eu.1	|- F/yph

Assertion

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

Proof of Theorem sb8eu

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . 5 |- F/w(ph <-> x = z)
2	1	sb8 2092	. . . 4 |- (A.x(ph <-> x = z) <-> A.w[w / x](ph <-> x = z))
3		sbbi 2071	. . . . . 6 |- ([w / x](ph <-> x = z) <-> ([w / x]ph <-> [w / x]x = z))
4		sb8eu.1	. . . . . . . 8 |- F/yph
5	4	nfsb 2109	. . . . . . 7 |- F/y[w / x]ph
6		equsb3 2102	. . . . . . . 8 |- ([w / x]x = z <-> w = z)
7		nfv 1619	. . . . . . . 8 |- F/y w = z
8	6, 7	nfxfr 1570	. . . . . . 7 |- F/y[w / x]x = z
9	5, 8	nfbi 1834	. . . . . 6 |- F/y([w / x]ph <-> [w / x]x = z)
10	3, 9	nfxfr 1570	. . . . 5 |- F/y[w / x](ph <-> x = z)
11		nfv 1619	. . . . 5 |- F/w[y / x](ph <-> x = z)
12		sbequ 2060	. . . . 5 |- (w = y -> ([w / x](ph <-> x = z) <-> [y / x](ph <-> x = z)))
13	10, 11, 12	cbval 1984	. . . 4 |- (A.w[w / x](ph <-> x = z) <-> A.y[y / x](ph <-> x = z))
14		equsb3 2102	. . . . . 6 |- ([y / x]x = z <-> y = z)
15	14	sblbis 2072	. . . . 5 |- ([y / x](ph <-> x = z) <-> ([y / x]ph <-> y = z))
16	15	albii 1566	. . . 4 |- (A.y[y / x](ph <-> x = z) <-> A.y([y / x]ph <-> y = z))
17	2, 13, 16	3bitri 262	. . 3 |- (A.x(ph <-> x = z) <-> A.y([y / x]ph <-> y = z))
18	17	exbii 1582	. 2 |- (E.zA.x(ph <-> x = z) <-> E.zA.y([y / x]ph <-> y = z))
19		df-eu 2208	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
20		df-eu 2208	. 2 |- (E!y[y / x]ph <-> E.zA.y([y / x]ph <-> y = z))
21	18, 19, 20	3bitr4i 268	1 |- (E!xph <-> E!y[y / x]ph)

Syntax hints:   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642  [wsb 1648  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  sb8mo  2223  cbveu  2224  eu1  2225  cbvreu  2834