Theorem altopeq2 35945

Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Assertion

Ref	Expression
altopeq2	⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)

Proof of Theorem altopeq2

Step	Hyp	Ref	Expression
1		eqid 2734	. 2 ⊢ 𝐶 = 𝐶
2		altopeq12 35943	. 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)
3	1, 2	mpan 690	1 ⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)

Syntax hints:   → wi 4   = wceq 1536  ⟪caltop 35937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-altop 35939

This theorem is referenced by:  sbcaltop  35962