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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 yφ
Assertion
Ref Expression
sb8eu (∃!xφ∃!y[y / x]φ)

Proof of Theorem sb8eu
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . 5 w(φx = z)
21sb8 2092 . . . 4 (x(φx = z) ↔ w[w / x](φx = z))
3 sbbi 2071 . . . . . 6 ([w / x](φx = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4 sb8eu.1 . . . . . . . 8 yφ
54nfsb 2109 . . . . . . 7 y[w / x]φ
6 equsb3 2102 . . . . . . . 8 ([w / x]x = zw = z)
7 nfv 1619 . . . . . . . 8 y w = z
86, 7nfxfr 1570 . . . . . . 7 y[w / x]x = z
95, 8nfbi 1834 . . . . . 6 y([w / x]φ ↔ [w / x]x = z)
103, 9nfxfr 1570 . . . . 5 y[w / x](φx = z)
11 nfv 1619 . . . . 5 w[y / x](φx = z)
12 sbequ 2060 . . . . 5 (w = y → ([w / x](φx = z) ↔ [y / x](φx = z)))
1310, 11, 12cbval 1984 . . . 4 (w[w / x](φx = z) ↔ y[y / x](φx = z))
14 equsb3 2102 . . . . . 6 ([y / x]x = zy = z)
1514sblbis 2072 . . . . 5 ([y / x](φx = z) ↔ ([y / x]φy = z))
1615albii 1566 . . . 4 (y[y / x](φx = z) ↔ y([y / x]φy = z))
172, 13, 163bitri 262 . . 3 (x(φx = z) ↔ y([y / x]φy = z))
1817exbii 1582 . 2 (zx(φx = z) ↔ zy([y / x]φy = z))
19 df-eu 2208 . 2 (∃!xφzx(φx = z))
20 df-eu 2208 . 2 (∃!y[y / x]φzy([y / x]φy = z))
2118, 19, 203bitr4i 268 1 (∃!xφ∃!y[y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648  ∃!weu 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  2833
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