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

Theorem morex 2719
Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
morex.1 B V
morex.2 (x = B → (φψ))
Assertion
Ref Expression
morex ((x A φ ∃*xφ) → (ψB A))
Distinct variable groups:   x,B   x,A   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem morex
StepHypRef Expression
1 df-rex 2306 . . . 4 (x A φx(x A φ))
2 exancom 1496 . . . 4 (x(x A φ) ↔ x(φ x A))
31, 2bitri 173 . . 3 (x A φx(φ x A))
4 nfmo1 1909 . . . . . 6 x∃*xφ
5 nfe1 1382 . . . . . 6 xx(φ x A)
64, 5nfan 1454 . . . . 5 x(∃*xφ x(φ x A))
7 mopick 1975 . . . . 5 ((∃*xφ x(φ x A)) → (φx A))
86, 7alrimi 1412 . . . 4 ((∃*xφ x(φ x A)) → x(φx A))
9 morex.1 . . . . 5 B V
10 morex.2 . . . . . 6 (x = B → (φψ))
11 eleq1 2097 . . . . . 6 (x = B → (x AB A))
1210, 11imbi12d 223 . . . . 5 (x = B → ((φx A) ↔ (ψB A)))
139, 12spcv 2640 . . . 4 (x(φx A) → (ψB A))
148, 13syl 14 . . 3 ((∃*xφ x(φ x A)) → (ψB A))
153, 14sylan2b 271 . 2 ((∃*xφ x A φ) → (ψB A))
1615ancoms 255 1 ((x A φ ∃*xφ) → (ψB A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  ∃*wmo 1898  wrex 2301  Vcvv 2551
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-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator