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Mirrors > Home > MPE Home > Th. List > soeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
Ref | Expression |
---|---|
soeq1 | ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poeq1 4962 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴)) | |
2 | breq 4585 | . . . . 5 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦)) | |
3 | biidd 251 | . . . . 5 ⊢ (𝑅 = 𝑆 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)) | |
4 | breq 4585 | . . . . 5 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥)) | |
5 | 2, 3, 4 | 3orbi123d 1390 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥))) |
6 | 5 | 2ralbidv 2972 | . . 3 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥))) |
7 | 1, 6 | anbi12d 743 | . 2 ⊢ (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥)))) |
8 | df-so 4960 | . 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
9 | df-so 4960 | . 2 ⊢ (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑆𝑥))) | |
10 | 7, 8, 9 | 3bitr4g 302 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∨ w3o 1030 = wceq 1475 ∀wral 2896 class class class wbr 4583 Po wpo 4957 Or wor 4958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-ex 1696 df-cleq 2603 df-clel 2606 df-ral 2901 df-br 4584 df-po 4959 df-so 4960 |
This theorem is referenced by: weeq1 5026 ltsopi 9589 cnso 14815 opsrtoslem2 19306 soeq12d 36626 |
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