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

Theorem eqeqan12rd 2053
 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 ((ψ φ) → (A = 𝐶B = 𝐷))

Proof of Theorem eqeqan12rd
StepHypRef Expression
1 eqeqan12rd.1 . . 3 (φA = B)
2 eqeqan12rd.2 . . 3 (ψ𝐶 = 𝐷)
31, 2eqeqan12d 2052 . 2 ((φ ψ) → (A = 𝐶B = 𝐷))
43ancoms 255 1 ((ψ φ) → (A = 𝐶B = 𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242 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-4 1397  ax-17 1416  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-cleq 2030 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator