Theorem altopeq1 36530

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 2762	. 2 ⊢ 𝐶 = 𝐶
2		altopeq12 36529	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
3	1, 2	mpan2 704	1 ⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟪caltop 36523

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-altop 36525

This theorem is used by:  sbcaltop  36548