Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqeqan1d Structured version   Visualization version   GIF version

Theorem eqeqan1d 34321
Description: Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
Hypothesis
Ref Expression
eqeqan1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eqeqan1d ((𝜑𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeqan1d
StepHypRef Expression
1 eqeqan1d.1 . 2 (𝜑𝐴 = 𝐵)
2 eqeq12 2818 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
31, 2sylan 571 1 ((𝜑𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-cleq 2798
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator