Theorem altopeq1 36413

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 2770	. 2 ⊢ 𝐶 = 𝐶
2		altopeq12 36412	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
3	1, 2	mpan2 703	1 ⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

Syntax hints:   → wi 4   = wceq 1568  ⟪caltop 36406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-un 3918  df-ss 3930  df-sn 4595  df-pr 4597  df-altop 36408

This theorem is referenced by:  sbcaltop  36431