Theorem ersymb 6811

Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
ersymb.1	|- ( ph -> R Er X )

Assertion

Ref	Expression
ersymb	|- ( ph -> ( A R B <-> B R A ) )

Proof of Theorem ersymb

Step	Hyp	Ref	Expression
1		ersymb.1	. . . 4 |- ( ph -> R Er X )
2	1	adantr 276	. . 3 |- ( ( ph /\ A R B ) -> R Er X )
3		simpr 110	. . 3 |- ( ( ph /\ A R B ) -> A R B )
4	2, 3	ersym 6809	. 2 |- ( ( ph /\ A R B ) -> B R A )
5	1	adantr 276	. . 3 |- ( ( ph /\ B R A ) -> R Er X )
6		simpr 110	. . 3 |- ( ( ph /\ B R A ) -> B R A )
7	5, 6	ersym 6809	. 2 |- ( ( ph /\ B R A ) -> A R B )
8	4, 7	impbida 604	1 |- ( ph -> ( A R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   class class class wbr 4125   Er wer 6794

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-er 6797

This theorem is referenced by:  ercnv  6818  erth  6843  erth2  6844  iinerm  6871  ensymb  7057