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Mirrors > Home > ILE Home > Th. List > poeq2 | GIF version |
Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
poeq2 | ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 3116 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | poss 4178 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Po 𝐴 → 𝑅 Po 𝐵)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 → 𝑅 Po 𝐵)) |
4 | eqimss 3115 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | poss 4178 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Po 𝐵 → 𝑅 Po 𝐴)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐵 → 𝑅 Po 𝐴)) |
7 | 3, 6 | impbid 128 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ⊆ wss 3035 Po wpo 4174 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-ral 2393 df-in 3041 df-ss 3048 df-po 4176 |
This theorem is referenced by: (None) |
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