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Mirrors > Home > MPE Home > Th. List > ersymb | Structured version Visualization version GIF version |
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ersymb.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
Ref | Expression |
---|---|
ersymb | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersymb.1 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝑅 Er 𝑋) |
3 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵) | |
4 | 2, 3 | ersym 8295 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴) |
5 | 1 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝑅 Er 𝑋) |
6 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴) | |
7 | 5, 6 | ersym 8295 | . 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵) |
8 | 4, 7 | impbida 799 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 class class class wbr 5058 Er wer 8280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-er 8283 |
This theorem is referenced by: ercnv 8304 erth 8332 erth2 8333 iiner 8363 ensymb 8551 |
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