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

Theorem eqvinc 2661
Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
eqvinc.1 A V
Assertion
Ref Expression
eqvinc (A = Bx(x = A x = B))
Distinct variable groups:   x,A   x,B

Proof of Theorem eqvinc
StepHypRef Expression
1 eqvinc.1 . . . . 5 A V
21isseti 2557 . . . 4 x x = A
3 ax-1 5 . . . . . 6 (x = A → (A = Bx = A))
4 eqtr 2054 . . . . . . 7 ((x = A A = B) → x = B)
54ex 108 . . . . . 6 (x = A → (A = Bx = B))
63, 5jca 290 . . . . 5 (x = A → ((A = Bx = A) (A = Bx = B)))
76eximi 1488 . . . 4 (x x = Ax((A = Bx = A) (A = Bx = B)))
8 pm3.43 534 . . . . 5 (((A = Bx = A) (A = Bx = B)) → (A = B → (x = A x = B)))
98eximi 1488 . . . 4 (x((A = Bx = A) (A = Bx = B)) → x(A = B → (x = A x = B)))
102, 7, 9mp2b 8 . . 3 x(A = B → (x = A x = B))
111019.37aiv 1562 . 2 (A = Bx(x = A x = B))
12 eqtr2 2055 . . 3 ((x = A x = B) → A = B)
1312exlimiv 1486 . 2 (x(x = A x = B) → A = B)
1411, 13impbii 117 1 (A = Bx(x = A x = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  eqvincf  2663
  Copyright terms: Public domain W3C validator