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Mirrors > Home > MPE Home > Th. List > Mathboxes > eel2131 | Structured version Visualization version GIF version |
Description: syl2an 493 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.) |
Ref | Expression |
---|---|
eel2131.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
eel2131.2 | ⊢ ((𝜑 ∧ 𝜃) → 𝜏) |
eel2131.3 | ⊢ ((𝜒 ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
eel2131 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eel2131.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | eel2131.2 | . . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏) | |
3 | eel2131.3 | . . 3 ⊢ ((𝜒 ∧ 𝜏) → 𝜂) | |
4 | 1, 2, 3 | syl2an 493 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) → 𝜂) |
5 | 4 | 3impdi 1373 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 df-3an 1033 |
This theorem is referenced by: (None) |
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