Theorem erth2 6639

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 6606	. 2 |- ( ph -> ( A R B <-> B R A ) )
3		erth2.2	. . . 4 |- ( ph -> B e. X )
4	1, 3	erth 6638	. . 3 |- ( ph -> ( B R A <-> [ B ] R = [ A ] R ) )
5		eqcom 2198	. . 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 2167   class class class wbr 4033   Er wer 6589   [cec 6590

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-er 6592  df-ec 6594

This theorem is referenced by:  qliftel  6674