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Theorem sb8eu 1891
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 1398 . . . . 5 w(φx = z)
21sb8 1714 . . . 4 (x(φx = z) ↔ w[w / x](φx = z))
3 sbbi 1811 . . . . . 6 ([w / x](φx = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4 sb8eu.1 . . . . . . . 8 yφ
54nfsb 1800 . . . . . . 7 y[w / x]φ
6 equsb3 1803 . . . . . . . 8 ([w / x]x = zw = z)
7 nfv 1398 . . . . . . . 8 y w = z
86, 7nfxfr 1339 . . . . . . 7 y[w / x]x = z
95, 8nfbi 1459 . . . . . 6 y([w / x]φ ↔ [w / x]x = z)
103, 9nfxfr 1339 . . . . 5 y[w / x](φx = z)
11 nfv 1398 . . . . 5 w[y / x](φx = z)
12 sbequ 1699 . . . . 5 (w = y → ([w / x](φx = z) ↔ [y / x](φx = z)))
1310, 11, 12cbval 1615 . . . 4 (w[w / x](φx = z) ↔ y[y / x](φx = z))
14 equsb3 1803 . . . . . 6 ([y / x]x = zy = z)
1514sblbis 1812 . . . . 5 ([y / x](φx = z) ↔ ([y / x]φy = z))
1615albii 1335 . . . 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 1474 . 2 (zx(φx = z) ↔ zy([y / x]φy = z))
19 df-eu 1881 . 2 (∃!xφzx(φx = z))
20 df-eu 1881 . 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 1224  wnf 1325  wex 1358  [wsb 1623  ∃!weu 1878
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881
This theorem is referenced by:  sb8mo  1892  nfeud  1894  nfeu  1897  cbveu  1902  cbvreu  2505  acexmid  5431
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