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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 446 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
3 | ibar 299 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
4 | ibar 299 | . . . 4 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒))) | |
5 | 3, 4 | bibi12d 234 | . . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))) |
6 | 5 | biimprcd 159 | . 2 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) |
7 | 2, 6 | impbii 125 | 1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: pm5.32i 450 biadani 602 xordidc 1388 cbvex2 1909 rabbi 2641 rabxfrd 4442 asymref 4984 rexrnmpt 5623 mpo2eqb 5943 |
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