Theorem erbr3b 32709

Description: Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.)

Assertion

Ref	Expression
erbr3b	⊢ ((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))

Proof of Theorem erbr3b

Step	Hyp	Ref	Expression
1		simpll 772	. . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝑅 Er 𝑋)
2		simplr 774	. . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐵)
3		simpr 485	. . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐶)
4	1, 2, 3	ertr3d 8652	. 2 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶)
5		simpll 772	. . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝑅 Er 𝑋)
6		simplr 774	. . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
7		simpr 485	. . 3 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
8	5, 6, 7	ertrd 8650	. 2 ⊢ (((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
9	4, 8	impbida 806	1 ⊢ ((𝑅 Er 𝑋 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   class class class wbr 5072   Er wer 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-er 8633

This theorem is referenced by: (None)