Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pclem6 | Structured version Visualization version GIF version |
Description: Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.) |
Ref | Expression |
---|---|
pclem6 | ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar 528 | . . 3 ⊢ (𝜓 → (¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑))) | |
2 | nbbn 384 | . . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) ↔ ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑))) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (𝜓 → ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑))) |
4 | 3 | con2i 139 | 1 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 |
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-an 396 |
This theorem is referenced by: rru 3717 nalset 5240 |
Copyright terms: Public domain | W3C validator |