Theorem ersymb 6527

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 6525	. 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 6525	. 2 |- ( ( ph /\ B R A ) -> A R B )
8	4, 7	impbida 591	1 |- ( ph -> ( A R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   class class class wbr 3989   Er wer 6510

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-er 6513

This theorem is referenced by:  ercnv  6534  erth  6557  erth2  6558  iinerm  6585  ensymb  6758