Theorem opkeq12i 4065

Description: Equality inference for ordered pairs. (The proof was shortened by Eric Schmidt, 4-Apr-2007.) (Contributed by NM, 16-Dec-2006.)

Hypotheses

Ref	Expression
opkeq1i.1	⊢ A = B
opkeq12i.2	⊢ C = D

Assertion

Ref	Expression
opkeq12i	⊢ ⟪A, C⟫ = ⟪B, D⟫

Proof of Theorem opkeq12i

Step	Hyp	Ref	Expression
1		opkeq1i.1	. 2 ⊢ A = B
2		opkeq12i.2	. 2 ⊢ C = D
3		opkeq12 4062	. 2 ⊢ ((A = B ∧ C = D) → ⟪A, C⟫ = ⟪B, D⟫)
4	1, 2, 3	mp2an 653	1 ⊢ ⟪A, C⟫ = ⟪B, D⟫

Syntax hints:   = wceq 1642  ⟪copk 4058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by: (None)