 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqeqan12dALT Structured version   Visualization version   GIF version

Theorem eqeqan12dALT 2622
 Description: Alternate proof of eqeqan12d 2621. This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eqeqan12d.1 (𝜑𝐴 = 𝐵)
eqeqan12d.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
eqeqan12dALT ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeqan12dALT
StepHypRef Expression
1 eqeqan12d.1 . 2 (𝜑𝐴 = 𝐵)
2 eqeqan12d.2 . 2 (𝜓𝐶 = 𝐷)
3 eqeq12 2618 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
41, 2, 3syl2an 492 1 ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 194   ∧ wa 382   = wceq 1474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-ext 2585 This theorem depends on definitions:  df-bi 195  df-an 384  df-cleq 2598 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator