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Mirrors > Home > MPE Home > Th. List > pm5.74 | Structured version Visualization version GIF version |
Description: Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) |
Ref | Expression |
---|---|
pm5.74 | ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 204 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
2 | 1 | imim3i 62 | . . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
3 | biimpr 209 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)) | |
4 | 3 | imim3i 62 | . . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓))) |
5 | 2, 4 | impbid 201 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
6 | biimp 204 | . . . 4 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | |
7 | 6 | pm2.86d 105 | . . 3 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒))) |
8 | biimpr 209 | . . . 4 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓))) | |
9 | 8 | pm2.86d 105 | . . 3 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜒 → 𝜓))) |
10 | 7, 9 | impbidd 199 | . 2 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) |
11 | 5, 10 | impbii 198 | 1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 |
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 196 |
This theorem is referenced by: pm5.74i 259 pm5.74ri 260 pm5.74d 261 pm5.74rd 262 bibi2d 331 pm5.32 666 orbidi 969 |
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