Theorem erth2 6634

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 6601	. 2 |- ( ph -> ( A R B <-> B R A ) )
3		erth2.2	. . . 4 |- ( ph -> B e. X )
4	1, 3	erth 6633	. . 3 |- ( ph -> ( B R A <-> [ B ] R = [ A ] R ) )
5		eqcom 2195	. . 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 1364   e. wcel 2164   class class class wbr 4029   Er wer 6584   [cec 6585

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-er 6587  df-ec 6589

This theorem is referenced by:  qliftel  6669