Theorem opeq2i 3822

Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Hypothesis

Ref	Expression
opeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
opeq2i	⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

Proof of Theorem opeq2i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		opeq2 3819	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

Syntax hints:   = wceq 1372  ⟨cop 3635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641

This theorem is referenced by:  fnressn  5760  fressnfv  5761  nqprlu  7642  suplocexpr  7820  addresr  7932  iseqvalcbv  10585  ressval2  12817  imasplusg  13058