Theorem ersymb 6443

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 274	. . 3 |- ( ( ph /\ A R B ) -> R Er X )
3		simpr 109	. . 3 |- ( ( ph /\ A R B ) -> A R B )
4	2, 3	ersym 6441	. 2 |- ( ( ph /\ A R B ) -> B R A )
5	1	adantr 274	. . 3 |- ( ( ph /\ B R A ) -> R Er X )
6		simpr 109	. . 3 |- ( ( ph /\ B R A ) -> B R A )
7	5, 6	ersym 6441	. 2 |- ( ( ph /\ B R A ) -> A R B )
8	4, 7	impbida 585	1 |- ( ph -> ( A R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   class class class wbr 3929   Er wer 6426

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-er 6429

This theorem is referenced by:  ercnv  6450  erth  6473  erth2  6474  iinerm  6501  ensymb  6674