Theorem opeq1i 4405

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

Hypothesis

Ref	Expression
opeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
opeq1i	⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩

Proof of Theorem opeq1i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		opeq1 4402	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩

Syntax hints:   = wceq 1483  ⟨cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  axi2m1  9980  s3tpop  13654  strlemor1OLD  15969  2strstr1  15986  2strop1  15988  grpbasex  15994  grpplusgx  15995  mat1dimelbas  20277  mat1dim0  20279  mat1dimid  20280  mat1dimscm  20281  mat1dimmul  20282  indistpsx  20814  setsiedg  25928  cusgrsize  26350  mapfzcons  37279