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Mirrors > Home > MPE Home > Th. List > pm5.32 | Structured version Visualization version GIF version |
Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
pm5.32 | ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notbi 308 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
2 | 1 | imbi2i 325 | . . 3 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))) |
3 | pm5.74 258 | . . 3 ⊢ ((𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) ↔ ((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒))) | |
4 | notbi 308 | . . 3 ⊢ (((𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒))) | |
5 | 2, 3, 4 | 3bitri 285 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒))) |
6 | df-an 385 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)) | |
7 | df-an 385 | . . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜑 → ¬ 𝜒)) | |
8 | 6, 7 | bibi12i 328 | . 2 ⊢ (((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) ↔ (¬ (𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜒))) |
9 | 5, 8 | bitr4i 266 | 1 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-an 385 |
This theorem is referenced by: pm5.32i 667 pm5.32d 669 xordi 935 rabbi 3097 rabxfrd 4815 asymref 5431 mpt22eqb 6667 cfilucfil4 22926 wl-ax11-lem8 32548 relexp0eq 37012 2sb5nd 37797 2sb5ndVD 38168 2sb5ndALT 38190 |
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