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Theorem ersymb 7376
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
StepHypRef Expression
1 ersymb.1 . . . 4  |-  ( ph  ->  R  Er  X )
21adantr 466 . . 3  |-  ( (
ph  /\  A R B )  ->  R  Er  X )
3 simpr 462 . . 3  |-  ( (
ph  /\  A R B )  ->  A R B )
42, 3ersym 7374 . 2  |-  ( (
ph  /\  A R B )  ->  B R A )
51adantr 466 . . 3  |-  ( (
ph  /\  B R A )  ->  R  Er  X )
6 simpr 462 . . 3  |-  ( (
ph  /\  B R A )  ->  B R A )
75, 6ersym 7374 . 2  |-  ( (
ph  /\  B R A )  ->  A R B )
84, 7impbida 840 1  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   class class class wbr 4417    Er wer 7359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853  df-er 7362
This theorem is referenced by:  ercnv  7383  erth  7407  erth2  7408  iiner  7434  ensymb  7615
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