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Mirrors > Home > ILE Home > Th. List > pm5.32 | GIF version |
Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.) |
Ref | Expression |
---|---|
pm5.32 | ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) | |
2 | 1 | pm5.32d 438 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
3 | ibar 295 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
4 | ibar 295 | . . . 4 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒))) | |
5 | 3, 4 | bibi12d 233 | . . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))) |
6 | 5 | biimprcd 158 | . 2 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) |
7 | 2, 6 | impbii 124 | 1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
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: pm5.32i 442 xordidc 1335 cbvex2 1845 rabbi 2544 rabxfrd 4291 asymref 4817 rexrnmpt 5442 mpt22eqb 5754 |
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