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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 111 | . . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜒 → 𝜃)) | |
3 | 1, 2 | syl6 29 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
4 | 3 | imdistand 421 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
5 | bi2 121 | . . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒)) | |
6 | 1, 5 | syl6 29 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) |
7 | 6 | imdistand 421 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜓 ∧ 𝜒))) |
8 | 4, 7 | impbid 120 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: pm5.32rd 424 pm5.32da 425 pm5.32 426 anbi2d 437 cbvex2 1797 cores 4824 isoini 5457 mpt2eq123 5564 genpassl 6622 genpassu 6623 fzind 8353 btwnz 8357 elfzm11 8953 |
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