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Mirrors > Home > ILE Home > Th. List > pm5.74 | 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 | bi1 116 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
2 | 1 | imim3i 60 | . . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
3 | bi2 128 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)) | |
4 | 3 | imim3i 60 | . . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓))) |
5 | 2, 4 | impbid 127 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
6 | bi1 116 | . . . 4 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | |
7 | 6 | pm2.86d 98 | . . 3 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒))) |
8 | bi2 128 | . . . 4 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓))) | |
9 | 8 | pm2.86d 98 | . . 3 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜒 → 𝜓))) |
10 | 7, 9 | impbidd 125 | . 2 ⊢ (((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) |
11 | 5, 10 | impbii 124 | 1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: pm5.74i 178 pm5.74ri 179 pm5.74d 180 pm5.74rd 181 bibi2d 230 |
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