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Theorem sb8eu 1910
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 1418 . . . . 5 w(φx = z)
21sb8 1733 . . . 4 (x(φx = z) ↔ w[w / x](φx = z))
3 sbbi 1830 . . . . . 6 ([w / x](φx = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4 sb8eu.1 . . . . . . . 8 yφ
54nfsb 1819 . . . . . . 7 y[w / x]φ
6 equsb3 1822 . . . . . . . 8 ([w / x]x = zw = z)
7 nfv 1418 . . . . . . . 8 y w = z
86, 7nfxfr 1360 . . . . . . 7 y[w / x]x = z
95, 8nfbi 1478 . . . . . 6 y([w / x]φ ↔ [w / x]x = z)
103, 9nfxfr 1360 . . . . 5 y[w / x](φx = z)
11 nfv 1418 . . . . 5 w[y / x](φx = z)
12 sbequ 1718 . . . . 5 (w = y → ([w / x](φx = z) ↔ [y / x](φx = z)))
1310, 11, 12cbval 1634 . . . 4 (w[w / x](φx = z) ↔ y[y / x](φx = z))
14 equsb3 1822 . . . . . 6 ([y / x]x = zy = z)
1514sblbis 1831 . . . . 5 ([y / x](φx = z) ↔ ([y / x]φy = z))
1615albii 1356 . . . 4 (y[y / x](φx = z) ↔ y([y / x]φy = z))
172, 13, 163bitri 195 . . 3 (x(φx = z) ↔ y([y / x]φy = z))
1817exbii 1493 . 2 (zx(φx = z) ↔ zy([y / x]φy = z))
19 df-eu 1900 . 2 (∃!xφzx(φx = z))
20 df-eu 1900 . 2 (∃!y[y / x]φzy([y / x]φy = z))
2118, 19, 203bitr4i 201 1 (∃!xφ∃!y[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240  wnf 1346  wex 1378  [wsb 1642  ∃!weu 1897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900
This theorem is referenced by:  sb8mo  1911  nfeud  1913  nfeu  1916  cbveu  1921  cbvreu  2525  acexmid  5454
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