Theorem erth2 6580

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 6549	. 2 |- ( ph -> ( A R B <-> B R A ) )
3		erth2.2	. . . 4 |- ( ph -> B e. X )
4	1, 3	erth 6579	. . 3 |- ( ph -> ( B R A <-> [ B ] R = [ A ] R ) )
5		eqcom 2179	. . 3 |- ( [ B ] R = [ A ] R <-> [ A ] R = [ B ] R )
6	4, 5	bitrdi 196	. 2 |- ( ph -> ( B R A <-> [ A ] R = [ B ] R ) )
7	2, 6	bitrd 188	1 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   class class class wbr 4004   Er wer 6532   [cec 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-er 6535  df-ec 6537

This theorem is referenced by:  qliftel  6615