Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqeq12i GIF version

Theorem eqeq12i 2151
 Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
eqeq12i.1 𝐴 = 𝐵
eqeq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
eqeq12i (𝐴 = 𝐶𝐵 = 𝐷)

Proof of Theorem eqeq12i
StepHypRef Expression
1 eqeq12i.1 . 2 𝐴 = 𝐵
2 eqeq12i.2 . 2 𝐶 = 𝐷
3 eqeq12 2150 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
41, 2, 3mp2an 422 1 (𝐴 = 𝐶𝐵 = 𝐷)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-cleq 2130 This theorem is referenced by:  rabbi  2606  sbceqg  3013  preqr2g  3689  preqr2  3691  otth  4159  rncoeq  4807  eqfnov  5870  mpo2eqb  5873  f1o2ndf1  6118  ecopovsym  6518  sq11i  10375  pwle2  13182
 Copyright terms: Public domain W3C validator