Theorem altopeq1 36185

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

Step	Hyp	Ref	Expression
1		eqid 2737	. 2 ⊢ 𝐶 = 𝐶
2		altopeq12 36184	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
3	1, 2	mpan2 692	1 ⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

Syntax hints:   → wi 4   = wceq 1542  ⟪caltop 36178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-sn 4583  df-pr 4585  df-altop 36180

This theorem is referenced by:  sbcaltop  36203