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Theorem opeq2d 3584
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
opeq2d (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Proof of Theorem opeq2d
StepHypRef Expression
1 opeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 opeq2 3578 . 2 (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
31, 2syl 14 1 (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by:  fundmen  6317  recexnq  6546  elreal2  6965  frecuzrdgrrn  9358  frec2uzrdg  9359  frecuzrdgsuc  9365  iseqeq2  9379  iseqeq3  9380  iseqval  9384
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