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Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > selconj | Structured version Visualization version GIF version |
Description: An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.) |
Ref | Expression |
---|---|
selconj.1 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Ref | Expression |
---|---|
selconj | ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜓 ∧ (𝜂 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | selconj.1 | . . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | |
2 | 1 | anbi2i 726 | . 2 ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒))) |
3 | an12 834 | . 2 ⊢ ((𝜓 ∧ (𝜂 ∧ 𝜒)) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒))) | |
4 | 2, 3 | bitr4i 266 | 1 ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜓 ∧ (𝜂 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ 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: (None) |
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