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Mirrors > Home > MPE Home > Th. List > Mathboxes > sylancl3 | Structured version Visualization version GIF version |
Description: Shortens 11 proofs by a total of around 150 bytes. (Contributed by BJ, 25-Apr-2019.) |
Ref | Expression |
---|---|
sylancl3.1 | ⊢ (𝜑 → 𝜓) |
sylancl3.2 | ⊢ 𝜒 |
sylancl3.3 | ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒)) |
Ref | Expression |
---|---|
sylancl3 | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylancl3.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | sylancl3.2 | . 2 ⊢ 𝜒 | |
3 | sylancl3.3 | . . 3 ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒)) | |
4 | 3 | biimpri 217 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
5 | 1, 2, 4 | sylancl 693 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → 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: bj-projval 32177 |
Copyright terms: Public domain | W3C validator |