Theorem erbr3b 29427

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

Assertion

Ref	Expression
erbr3b	|- ( ( R Er X /\ A R B ) -> ( A R C <-> B R C ) )

Proof of Theorem erbr3b

Step	Hyp	Ref	Expression
1		simpll 790	. . 3 |- ( ( ( R Er X /\ A R B ) /\ A R C ) -> R Er X )
2		simplr 792	. . 3 |- ( ( ( R Er X /\ A R B ) /\ A R C ) -> A R B )
3		simpr 477	. . 3 |- ( ( ( R Er X /\ A R B ) /\ A R C ) -> A R C )
4	1, 2, 3	ertr3d 7760	. 2 |- ( ( ( R Er X /\ A R B ) /\ A R C ) -> B R C )
5		simpll 790	. . 3 |- ( ( ( R Er X /\ A R B ) /\ B R C ) -> R Er X )
6		simplr 792	. . 3 |- ( ( ( R Er X /\ A R B ) /\ B R C ) -> A R B )
7		simpr 477	. . 3 |- ( ( ( R Er X /\ A R B ) /\ B R C ) -> B R C )
8	5, 6, 7	ertrd 7758	. 2 |- ( ( ( R Er X /\ A R B ) /\ B R C ) -> A R C )
9	4, 8	impbida 877	1 |- ( ( R Er X /\ A R B ) -> ( A R C <-> B R C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   class class class wbr 4653   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-er 7742

This theorem is referenced by: (None)