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Mirrors > Home > ILE Home > Th. List > pm5.32d | GIF version |
Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.) |
Ref | Expression |
---|---|
pm5.32d.1 | ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) |
Ref | Expression |
---|---|
pm5.32d | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.32d.1 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) | |
2 | bi1 116 | . . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜒 → 𝜃)) | |
3 | 1, 2 | syl6 33 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
4 | 3 | imdistand 435 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
5 | bi2 128 | . . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒)) | |
6 | 1, 5 | syl6 33 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) |
7 | 6 | imdistand 435 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜓 ∧ 𝜒))) |
8 | 4, 7 | impbid 127 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ 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: pm5.32rd 438 pm5.32da 439 pm5.32 440 anbi2d 451 cbvex2 1838 cores 4844 isoini 5477 mpt2eq123 5584 genpassl 6714 genpassu 6715 fzind 8462 btwnz 8466 elfzm11 9108 isprm2 10499 isprm3 10500 |
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