New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  mo2icl GIF version

Theorem mo2icl 3015
 Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
Assertion
Ref Expression
mo2icl (x(φx = A) → ∃*xφ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem mo2icl
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2362 . . . . . 6 (y = A → (x = yx = A))
21imbi2d 307 . . . . 5 (y = A → ((φx = y) ↔ (φx = A)))
32albidv 1625 . . . 4 (y = A → (x(φx = y) ↔ x(φx = A)))
43imbi1d 308 . . 3 (y = A → ((x(φx = y) → ∃*xφ) ↔ (x(φx = A) → ∃*xφ)))
5 19.8a 1756 . . . 4 (x(φx = y) → yx(φx = y))
6 nfv 1619 . . . . 5 yφ
76mo2 2233 . . . 4 (∃*xφyx(φx = y))
85, 7sylibr 203 . . 3 (x(φx = y) → ∃*xφ)
94, 8vtoclg 2914 . 2 (A V → (x(φx = A) → ∃*xφ))
10 vex 2862 . . . . . . 7 x V
11 eleq1 2413 . . . . . . 7 (x = A → (x V ↔ A V))
1210, 11mpbii 202 . . . . . 6 (x = AA V)
1312imim2i 13 . . . . 5 ((φx = A) → (φA V))
1413con3rr3 128 . . . 4 A V → ((φx = A) → ¬ φ))
1514alimdv 1621 . . 3 A V → (x(φx = A) → x ¬ φ))
16 alnex 1543 . . . 4 (x ¬ φ ↔ ¬ xφ)
17 exmo 2249 . . . . 5 (xφ ∃*xφ)
1817ori 364 . . . 4 xφ∃*xφ)
1916, 18sylbi 187 . . 3 (x ¬ φ∃*xφ)
2015, 19syl6 29 . 2 A V → (x(φx = A) → ∃*xφ))
219, 20pm2.61i 156 1 (x(φx = A) → ∃*xφ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  Vcvv 2859 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-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: (None)
 Copyright terms: Public domain W3C validator