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

Theorem moim 1961
 Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
Assertion
Ref Expression
moim (x(φψ) → (∃*xψ∃*xφ))

Proof of Theorem moim
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfa1 1431 . . 3 xx(φψ)
2 ax-4 1397 . . . . . 6 (x(φψ) → (φψ))
3 spsbim 1721 . . . . . 6 (x(φψ) → ([y / x]φ → [y / x]ψ))
42, 3anim12d 318 . . . . 5 (x(φψ) → ((φ [y / x]φ) → (ψ [y / x]ψ)))
54imim1d 69 . . . 4 (x(φψ) → (((ψ [y / x]ψ) → x = y) → ((φ [y / x]φ) → x = y)))
65alimdv 1756 . . 3 (x(φψ) → (y((ψ [y / x]ψ) → x = y) → y((φ [y / x]φ) → x = y)))
71, 6alimd 1411 . 2 (x(φψ) → (xy((ψ [y / x]ψ) → x = y) → xy((φ [y / x]φ) → x = y)))
8 ax-17 1416 . . 3 (ψyψ)
98mo3h 1950 . 2 (∃*xψxy((ψ [y / x]ψ) → x = y))
10 ax-17 1416 . . 3 (φyφ)
1110mo3h 1950 . 2 (∃*xφxy((φ [y / x]φ) → x = y))
127, 9, 113imtr4g 194 1 (x(φψ) → (∃*xψ∃*xφ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  [wsb 1642  ∃*wmo 1898 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901 This theorem is referenced by:  moimi  1962  euimmo  1964  moexexdc  1981  euexex  1982  rmoim  2734  rmoimi2  2736  disjss1  3742  reusv1  4156  funmo  4860
 Copyright terms: Public domain W3C validator