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Theorem mob2 2694
 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 892 . . 3 ((A B ∃*xφ φ) → φ)
2 moi2.1 . . 3 (x = A → (φψ))
31, 2syl5ibcom 144 . 2 ((A B ∃*xφ φ) → (x = Aψ))
4 nfs1v 1793 . . . . . . . 8 x[y / x]φ
5 sbequ12 1632 . . . . . . . 8 (x = y → (φ ↔ [y / x]φ))
64, 5mo4f 1938 . . . . . . 7 (∃*xφxy((φ [y / x]φ) → x = y))
7 sp 1378 . . . . . . 7 (xy((φ [y / x]φ) → x = y) → y((φ [y / x]φ) → x = y))
86, 7sylbi 114 . . . . . 6 (∃*xφy((φ [y / x]φ) → x = y))
9 nfv 1398 . . . . . . . . . 10 xψ
109, 2sbhypf 2576 . . . . . . . . 9 (y = A → ([y / x]φψ))
1110anbi2d 440 . . . . . . . 8 (y = A → ((φ [y / x]φ) ↔ (φ ψ)))
12 eqeq2 2027 . . . . . . . 8 (y = A → (x = yx = A))
1311, 12imbi12d 223 . . . . . . 7 (y = A → (((φ [y / x]φ) → x = y) ↔ ((φ ψ) → x = A)))
1413spcgv 2613 . . . . . 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 115 . . . 4 ((A B ∃*xφ) → ((φ ψ) → x = A))
1716expd 245 . . 3 ((A B ∃*xφ) → (φ → (ψx = A)))
18173impia 1085 . 2 ((A B ∃*xφ φ) → (ψx = A))
193, 18impbid 120 1 ((A B ∃*xφ φ) → (x = Aψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 871  ∀wal 1224   = wceq 1226   ∈ wcel 1370  [wsb 1623  ∃*wmo 1879 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  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533 This theorem is referenced by:  moi2  2695  mob  2696
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