Theorem ersymb 6636

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 6634	. 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 6634	. 2 |- ( ( ph /\ B R A ) -> A R B )
8	4, 7	impbida 596	1 |- ( ph -> ( A R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   class class class wbr 4045   Er wer 6619

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-cnv 4684  df-er 6622

This theorem is referenced by:  ercnv  6643  erth  6668  erth2  6669  iinerm  6696  ensymb  6874