Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rb-imdf | Structured version Visualization version GIF version |
Description: The definition of implication, in terms of ∨ and ¬. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rb-imdf | ⊢ ¬ (¬ (¬ (𝜑 → 𝜓) ∨ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imor 853 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | |
2 | rb-bijust 1757 | . 2 ⊢ (((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) ↔ ¬ (¬ (¬ (𝜑 → 𝜓) ∨ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜑 → 𝜓)))) | |
3 | 1, 2 | mpbi 233 | 1 ⊢ ¬ (¬ (¬ (𝜑 → 𝜓) ∨ (¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 847 |
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 210 df-or 848 |
This theorem is referenced by: re1axmp 1772 re2luk1 1773 re2luk2 1774 re2luk3 1775 |
Copyright terms: Public domain | W3C validator |