Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pm4.72 | Structured version Visualization version GIF version |
Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.) |
Ref | Expression |
---|---|
pm4.72 | ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 865 | . . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
2 | pm2.621 896 | . . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓)) | |
3 | 1, 2 | impbid2 225 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜓 ↔ (𝜑 ∨ 𝜓))) |
4 | orc 864 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
5 | biimpr 219 | . . 3 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → ((𝜑 ∨ 𝜓) → 𝜓)) | |
6 | 4, 5 | syl5 34 | . 2 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → (𝜑 → 𝜓)) |
7 | 3, 6 | impbii 208 | 1 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 844 |
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 206 df-or 845 |
This theorem is referenced by: bigolden 1024 cadan 1611 ssequn1 4114 ssunsn2 4760 vtxd0nedgb 27855 bj-consensusALT 34760 wl-ifpimpr 35637 elpaddn0 37814 |
Copyright terms: Public domain | W3C validator |