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Theorem sbmo 2234
 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 sbex 2128 . . 3 ([y / x]wz(φz = w) ↔ w[y / x]z(φz = w))
2 nfv 1619 . . . . . 6 x z = w
32sblim 2068 . . . . 5 ([y / x](φz = w) ↔ ([y / x]φz = w))
43sbalv 2129 . . . 4 ([y / x]z(φz = w) ↔ z([y / x]φz = w))
54exbii 1582 . . 3 (w[y / x]z(φz = w) ↔ wz([y / x]φz = w))
61, 5bitri 240 . 2 ([y / x]wz(φz = w) ↔ wz([y / x]φz = w))
7 nfv 1619 . . . 4 wφ
87mo2 2233 . . 3 (∃*zφwz(φz = w))
98sbbii 1653 . 2 ([y / x]∃*zφ ↔ [y / x]wz(φz = w))
10 nfv 1619 . . 3 w[y / x]φ
1110mo2 2233 . 2 (∃*z[y / x]φwz([y / x]φz = w))
126, 9, 113bitr4i 268 1 ([y / x]∃*zφ∃*z[y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃*wmo 2205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by: (None)
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