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Theorem ersymb 6304
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
Assertion
Ref Expression
ersymb (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem ersymb
StepHypRef Expression
1 ersymb.1 . . . 4 (𝜑𝑅 Er 𝑋)
21adantr 270 . . 3 ((𝜑𝐴𝑅𝐵) → 𝑅 Er 𝑋)
3 simpr 108 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3ersym 6302 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 270 . . 3 ((𝜑𝐵𝑅𝐴) → 𝑅 Er 𝑋)
6 simpr 108 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6ersym 6302 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 563 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   class class class wbr 3845   Er wer 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-er 6290
This theorem is referenced by:  ercnv  6311  erth  6334  erth2  6335  iinerm  6362  ensymb  6495
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