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Theorem ersymb 8156
 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 481 . . 3 ((𝜑𝐴𝑅𝐵) → 𝑅 Er 𝑋)
3 simpr 485 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3ersym 8154 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 481 . . 3 ((𝜑𝐵𝑅𝐴) → 𝑅 Er 𝑋)
6 simpr 485 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6ersym 8154 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 797 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   class class class wbr 4964   Er wer 8139 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-br 4965  df-opab 5027  df-xp 5452  df-rel 5453  df-cnv 5454  df-er 8142 This theorem is referenced by:  ercnv  8163  erth  8191  erth2  8192  iiner  8222  ensymb  8408
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