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Theorem erbr3b 32629
Description: Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.)
Assertion
Ref Expression
erbr3b ((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))

Proof of Theorem erbr3b
StepHypRef Expression
1 simpll 767 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝑅 Er 𝑋)
2 simplr 769 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐵)
3 simpr 484 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐶)
41, 2, 3ertr3d 8763 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶)
5 simpll 767 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝑅 Er 𝑋)
6 simplr 769 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
7 simpr 484 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
85, 6, 7ertrd 8761 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
94, 8impbida 801 1 ((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   class class class wbr 5143   Er wer 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-er 8745
This theorem is referenced by: (None)
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