Theorem altopeq1 36007

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 2731	. 2 ⊢ 𝐶 = 𝐶
2		altopeq12 36006	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
3	1, 2	mpan2 691	1 ⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

Syntax hints:   → wi 4   = wceq 1541  ⟪caltop 36000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-sn 4574  df-pr 4576  df-altop 36002

This theorem is referenced by:  sbcaltop  36025