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Theorem erbr3b 29555
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 805 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝑅 Er 𝑋)
2 simplr 807 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐵)
3 simpr 476 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐶)
41, 2, 3ertr3d 7805 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶)
5 simpll 805 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝑅 Er 𝑋)
6 simplr 807 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
7 simpr 476 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
85, 6, 7ertrd 7803 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
94, 8impbida 895 1 ((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   class class class wbr 4685   Er wer 7784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-er 7787
This theorem is referenced by: (None)
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