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Theorem erbr3b 32636
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 8761 . 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 8759 . 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 5147   Er wer 8740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-er 8743
This theorem is referenced by: (None)
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