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

Theorem sbmo 1941
 Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
sbmo ([y / x]∃*zφ∃*z[y / x]φ)
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbmo
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1402 . . . . . 6 x z = w
21sblim 1813 . . . . 5 ([y / x]((φ [w / z]φ) → z = w) ↔ ([y / x](φ [w / z]φ) → z = w))
3 sban 1811 . . . . . 6 ([y / x](φ [w / z]φ) ↔ ([y / x]φ [y / x][w / z]φ))
43imbi1i 227 . . . . 5 (([y / x](φ [w / z]φ) → z = w) ↔ (([y / x]φ [y / x][w / z]φ) → z = w))
5 sbcom2 1845 . . . . . . 7 ([y / x][w / z]φ ↔ [w / z][y / x]φ)
65anbi2i 433 . . . . . 6 (([y / x]φ [y / x][w / z]φ) ↔ ([y / x]φ [w / z][y / x]φ))
76imbi1i 227 . . . . 5 ((([y / x]φ [y / x][w / z]φ) → z = w) ↔ (([y / x]φ [w / z][y / x]φ) → z = w))
82, 4, 73bitri 195 . . . 4 ([y / x]((φ [w / z]φ) → z = w) ↔ (([y / x]φ [w / z][y / x]φ) → z = w))
98sbalv 1863 . . 3 ([y / x]w((φ [w / z]φ) → z = w) ↔ w(([y / x]φ [w / z][y / x]φ) → z = w))
109sbalv 1863 . 2 ([y / x]zw((φ [w / z]φ) → z = w) ↔ zw(([y / x]φ [w / z][y / x]φ) → z = w))
11 nfv 1402 . . . 4 wφ
1211mo3 1936 . . 3 (∃*zφzw((φ [w / z]φ) → z = w))
1312sbbii 1630 . 2 ([y / x]∃*zφ ↔ [y / x]zw((φ [w / z]φ) → z = w))
14 nfv 1402 . . 3 w[y / x]φ
1514mo3 1936 . 2 (∃*z[y / x]φzw(([y / x]φ [w / z][y / x]φ) → z = w))
1610, 13, 153bitr4i 201 1 ([y / x]∃*zφ∃*z[y / x]φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228  [wsb 1627  ∃*wmo 1883 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator