Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  altopeq1 Structured version   Visualization version   GIF version

Theorem altopeq1 35592
Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopeq1 (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

Proof of Theorem altopeq1
StepHypRef Expression
1 eqid 2728 . 2 𝐶 = 𝐶
2 altopeq12 35591 . 2 ((𝐴 = 𝐵𝐶 = 𝐶) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
31, 2mpan2 689 1 (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  caltop 35585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-sn 4633  df-pr 4635  df-altop 35587
This theorem is referenced by:  sbcaltop  35610
  Copyright terms: Public domain W3C validator