ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqeqan12rd GIF version

Theorem eqeqan12rd 2072
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
Hypotheses
Ref Expression
eqeqan12rd.1 (𝜑𝐴 = 𝐵)
eqeqan12rd.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
eqeqan12rd ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeqan12rd
StepHypRef Expression
1 eqeqan12rd.1 . . 3 (𝜑𝐴 = 𝐵)
2 eqeqan12rd.2 . . 3 (𝜓𝐶 = 𝐷)
31, 2eqeqan12d 2071 . 2 ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
43ancoms 259 1 ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator