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Theorem opeq1i 4596
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
StepHypRef Expression
1 opeq1i.1 . 2 𝐴 = 𝐵
2 opeq1 4593 . 2 (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
31, 2ax-mp 5 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  cop 4374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375
This theorem is referenced by:  axi2m1  10268  s3tpop  13994  2strstr1  16307  2strop1  16309  grpbasex  16315  grpplusgx  16316  mat1dimelbas  20603  mat1dim0  20605  mat1dimid  20606  mat1dimscm  20607  mat1dimmul  20608  indistpsx  21143  setsiedg  26271  cusgrsize  26704  mapfzcons  38065
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