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Theorem mob2 3016
 Description: Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
Hypothesis
Ref Expression
moi2.1 (x = A → (φψ))
Assertion
Ref Expression
mob2 ((A B ∃*xφ φ) → (x = Aψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem mob2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . 3 ((A B ∃*xφ φ) → φ)
2 moi2.1 . . 3 (x = A → (φψ))
31, 2syl5ibcom 211 . 2 ((A B ∃*xφ φ) → (x = Aψ))
4 nfs1v 2106 . . . . . . . 8 x[y / x]φ
5 sbequ12 1919 . . . . . . . 8 (x = y → (φ ↔ [y / x]φ))
64, 5mo4f 2236 . . . . . . 7 (∃*xφxy((φ [y / x]φ) → x = y))
7 sp 1747 . . . . . . 7 (xy((φ [y / x]φ) → x = y) → y((φ [y / x]φ) → x = y))
86, 7sylbi 187 . . . . . 6 (∃*xφy((φ [y / x]φ) → x = y))
9 nfv 1619 . . . . . . . . . 10 xψ
109, 2sbhypf 2904 . . . . . . . . 9 (y = A → ([y / x]φψ))
1110anbi2d 684 . . . . . . . 8 (y = A → ((φ [y / x]φ) ↔ (φ ψ)))
12 eqeq2 2362 . . . . . . . 8 (y = A → (x = yx = A))
1311, 12imbi12d 311 . . . . . . 7 (y = A → (((φ [y / x]φ) → x = y) ↔ ((φ ψ) → x = A)))
1413spcgv 2939 . . . . . 6 (A B → (y((φ [y / x]φ) → x = y) → ((φ ψ) → x = A)))
158, 14syl5 28 . . . . 5 (A B → (∃*xφ → ((φ ψ) → x = A)))
1615imp 418 . . . 4 ((A B ∃*xφ) → ((φ ψ) → x = A))
1716exp3a 425 . . 3 ((A B ∃*xφ) → (φ → (ψx = A)))
18173impia 1148 . 2 ((A B ∃*xφ φ) → (ψx = A))
193, 18impbid 183 1 ((A B ∃*xφ φ) → (x = Aψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃*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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by:  moi2  3017  mob  3018
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