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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 4039 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | poss 5592 | . . 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 4038 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | poss 5592 | . . 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 211 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ⊆ wss 3947 Po wpo 5588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-v 3473 df-in 3954 df-ss 3964 df-po 5590 |
This theorem is referenced by: posn 5763 dfpo2 6300 frfi 9313 ipo0 43886 |
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