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Theorem erbr3b 32101
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 765 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝑅 Er 𝑋)
2 simplr 767 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐵)
3 simpr 485 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐶)
41, 2, 3ertr3d 8723 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶)
5 simpll 765 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝑅 Er 𝑋)
6 simplr 767 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
7 simpr 485 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
85, 6, 7ertrd 8721 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
94, 8impbida 799 1 ((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   class class class wbr 5148   Er wer 8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-er 8705
This theorem is referenced by: (None)
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