Theorem ersymb 8686

Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
ersymb.1	⊢ (𝜑 → 𝑅 Er 𝑋)

Assertion

Ref	Expression
ersymb	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

Proof of Theorem ersymb

Step	Hyp	Ref	Expression
1		ersymb.1	. . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋)
2	1	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝑅 Er 𝑋)
3		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4	2, 3	ersym 8684	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
5	1	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝑅 Er 𝑋)
6		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴)
7	5, 6	ersym 8684	. 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵)
8	4, 7	impbida 810	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   class class class wbr 5099   Er wer 8668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-er 8671

This theorem is referenced by:  ercnv  8693  erth  8726  erth2  8727  iiner  8764  ensymb  8977