ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mosubt Structured version   GIF version

Theorem mosubt 2682
Description: "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
Assertion
Ref Expression
mosubt (y∃*xφ∃*xy(y = A φ))
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem mosubt
StepHypRef Expression
1 eueq 2676 . . . . . 6 (A V ↔ ∃!y y = A)
2 isset 2526 . . . . . 6 (A V ↔ y y = A)
31, 2bitr3i 175 . . . . 5 (∃!y y = Ay y = A)
4 nfv 1390 . . . . . 6 x y = A
54euexex 1954 . . . . 5 ((∃!y y = A y∃*xφ) → ∃*xy(y = A φ))
63, 5sylanbr 269 . . . 4 ((y y = A y∃*xφ) → ∃*xy(y = A φ))
76expcom 109 . . 3 (y∃*xφ → (y y = A∃*xy(y = A φ)))
8 moanimv 1944 . . 3 (∃*x(y y = A y(y = A φ)) ↔ (y y = A∃*xy(y = A φ)))
97, 8sylibr 137 . 2 (y∃*xφ∃*x(y y = A y(y = A φ)))
10 ax-ia1 99 . . . . 5 ((y = A φ) → y = A)
1110eximi 1460 . . . 4 (y(y = A φ) → y y = A)
1211ancri 307 . . 3 (y(y = A φ) → (y y = A y(y = A φ)))
1312moimi 1934 . 2 (∃*x(y y = A y(y = A φ)) → ∃*xy(y = A φ))
149, 13syl 14 1 (y∃*xφ∃*xy(y = A φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1217   = wceq 1219  wex 1350   wcel 1362  ∃!weu 1869  ∃*wmo 1870  Vcvv 2522
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 613  ax-5 1305  ax-7 1306  ax-gen 1307  ax-ie1 1351  ax-ie2 1352  ax-8 1364  ax-10 1365  ax-11 1366  ax-i12 1367  ax-bnd 1368  ax-4 1369  ax-17 1388  ax-i9 1392  ax-ial 1396  ax-i5r 1397  ax-ext 1991
This theorem depends on definitions:  df-bi 110  df-tru 1222  df-nf 1319  df-sb 1615  df-eu 1872  df-mo 1873  df-clab 1996  df-cleq 2002  df-clel 2005  df-v 2524
This theorem is referenced by:  mosub  2683
  Copyright terms: Public domain W3C validator