Theorem erth2 6467

Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Hypotheses

Ref	Expression
erth2.1	|- ( ph -> R Er X )
erth2.2	|- ( ph -> B e. X )

Assertion

Ref	Expression
erth2	|- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Proof of Theorem erth2

Step	Hyp	Ref	Expression
1		erth2.1	. . 3 |- ( ph -> R Er X )
2	1	ersymb 6436	. 2 |- ( ph -> ( A R B <-> B R A ) )
3		erth2.2	. . . 4 |- ( ph -> B e. X )
4	1, 3	erth 6466	. . 3 |- ( ph -> ( B R A <-> [ B ] R = [ A ] R ) )
5		eqcom 2139	. . 3 |- ( [ B ] R = [ A ] R <-> [ A ] R = [ B ] R )
6	4, 5	syl6bb 195	. 2 |- ( ph -> ( B R A <-> [ A ] R = [ B ] R ) )
7	2, 6	bitrd 187	1 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   class class class wbr 3924   Er wer 6419   [cec 6420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-er 6422  df-ec 6424

This theorem is referenced by:  qliftel  6502