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Mirrors > Home > MPE Home > Th. List > pm5.501 | Structured version Visualization version GIF version |
Description: Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
pm5.501 | ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.1im 266 | . 2 ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓))) | |
2 | biimp 218 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
3 | 2 | com12 32 | . 2 ⊢ (𝜑 → ((𝜑 ↔ 𝜓) → 𝜓)) |
4 | 1, 3 | impbid 215 | 1 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 |
This theorem is referenced by: ibib 371 ibibr 372 nbn2 374 pm5.18 386 biass 389 pm5.1 824 vn0 4270 sadadd2lem2 16042 isclo 22016 nrmmetd 23504 bj-bibibi 34539 |
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