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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 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝑅 Er 𝑋) |
3 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵) | |
4 | 2, 3 | ersym 8154 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴) |
5 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝑅 Er 𝑋) |
6 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴) | |
7 | 5, 6 | ersym 8154 | . 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵) |
8 | 4, 7 | impbida 797 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
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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