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Mirrors > Home > MPE Home > Th. List > poeq2 | Structured version Visualization version 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 4055 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | poss 5599 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Po 𝐴 → 𝑅 Po 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 → 𝑅 Po 𝐵)) |
4 | eqimss 4054 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | poss 5599 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Po 𝐵 → 𝑅 Po 𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐵 → 𝑅 Po 𝐴)) |
7 | 3, 6 | impbid 212 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ⊆ wss 3963 Po wpo 5595 |
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 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ral 3060 df-ss 3980 df-po 5597 |
This theorem is referenced by: poeq12d 5602 posn 5774 dfpo2 6318 frfi 9319 ipo0 44445 |
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