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Theorem ersymb 6056
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 261 . . 3 ((φ A𝑅B) → 𝑅 Er 𝑋)
3 simpr 103 . . 3 ((φ A𝑅B) → A𝑅B)
42, 3ersym 6054 . 2 ((φ A𝑅B) → B𝑅A)
51adantr 261 . . 3 ((φ B𝑅A) → 𝑅 Er 𝑋)
6 simpr 103 . . 3 ((φ B𝑅A) → B𝑅A)
75, 6ersym 6054 . 2 ((φ B𝑅A) → A𝑅B)
84, 7impbida 528 1 (φ → (A𝑅BB𝑅A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   class class class wbr 3755   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-er 6042
This theorem is referenced by:  ercnv  6063  erth  6086  erth2  6087  iinerm  6114  ensymb  6196
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