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Theorem sbmo 1956
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 1418 . . . . . 6 x z = w
21sblim 1828 . . . . 5 ([y / x]((φ [w / z]φ) → z = w) ↔ ([y / x](φ [w / z]φ) → z = w))
3 sban 1826 . . . . . 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 1860 . . . . . . 7 ([y / x][w / z]φ ↔ [w / z][y / x]φ)
65anbi2i 430 . . . . . 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 1878 . . 3 ([y / x]w((φ [w / z]φ) → z = w) ↔ w(([y / x]φ [w / z][y / x]φ) → z = w))
109sbalv 1878 . 2 ([y / x]zw((φ [w / z]φ) → z = w) ↔ zw(([y / x]φ [w / z][y / x]φ) → z = w))
11 nfv 1418 . . . 4 wφ
1211mo3 1951 . . 3 (∃*zφzw((φ [w / z]φ) → z = w))
1312sbbii 1645 . 2 ([y / x]∃*zφ ↔ [y / x]zw((φ [w / z]φ) → z = w))
14 nfv 1418 . . 3 w[y / x]φ
1514mo3 1951 . 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 1240   = wceq 1242  [wsb 1642  ∃*wmo 1898
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-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by: (None)
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