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Theorem opeq12i 3545
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
Hypotheses
Ref Expression
opeq1i.1 A = B
opeq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
opeq12i A, 𝐶⟩ = ⟨B, 𝐷

Proof of Theorem opeq12i
StepHypRef Expression
1 opeq1i.1 . 2 A = B
2 opeq12i.2 . 2 𝐶 = 𝐷
3 opeq12 3542 . 2 ((A = B 𝐶 = 𝐷) → ⟨A, 𝐶⟩ = ⟨B, 𝐷⟩)
41, 2, 3mp2an 402 1 A, 𝐶⟩ = ⟨B, 𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1242  cop 3370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  addpinq1  6447  genipv  6492  ltexpri  6587  recexpr  6610  cauappcvgprlemladdru  6628  cauappcvgprlemladdrl  6629  cauappcvgpr  6634  caucvgprlemcl  6647  caucvgprlemladdrl  6649  caucvgpr  6653  pitonnlem1  6741  axi2m1  6759
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