Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 423 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
3 | ibar 285 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
4 | ibar 285 | . . . 4 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒))) | |
5 | 3, 4 | bibi12d 224 | . . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))) |
6 | 5 | biimprcd 149 | . 2 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) |
7 | 2, 6 | impbii 117 | 1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
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: pm5.32i 427 xordidc 1290 cbvex2 1797 rabbi 2487 rabxfrd 4201 asymref 4710 rexrnmpt 5310 mpt22eqb 5610 |
Copyright terms: Public domain | W3C validator |