Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8euh Structured version   GIF version

Theorem sb8euh 1901
 Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)
Hypothesis
Ref Expression
sb8euh.1 (φyφ)
Assertion
Ref Expression
sb8euh (∃!xφ∃!y[y / x]φ)

Proof of Theorem sb8euh
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-17 1396 . . . . 5 ((φx = z) → w(φx = z))
21sb8h 1712 . . . 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 sb8euh.1 . . . . . . . 8 (φyφ)
54hbsb 1801 . . . . . . 7 ([w / x]φy[w / x]φ)
6 equsb3 1803 . . . . . . . 8 ([w / x]x = zw = z)
7 ax-17 1396 . . . . . . . 8 (w = zy w = z)
86, 7hbxfrbi 1337 . . . . . . 7 ([w / x]x = zy[w / x]x = z)
95, 8hbbi 1418 . . . . . 6 (([w / x]φ ↔ [w / x]x = z) → y([w / x]φ ↔ [w / x]x = z))
103, 9hbxfrbi 1337 . . . . 5 ([w / x](φx = z) → y[w / x](φx = z))
11 ax-17 1396 . . . . 5 ([y / x](φx = z) → w[y / x](φx = z))
12 sbequ 1699 . . . . 5 (w = y → ([w / x](φx = z) ↔ [y / x](φx = z)))
1310, 11, 12cbvalh 1614 . . . 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:   → wi 4   ↔ wb 98  ∀wal 1224  ∃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-nf 1326  df-sb 1624  df-eu 1881 This theorem is referenced by:  eu1  1903
 Copyright terms: Public domain W3C validator