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Mirrors > Home > ILE Home > Th. List > bianass | GIF version |
Description: An inference to merge two lists of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.) |
Ref | Expression |
---|---|
bianass.1 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Ref | Expression |
---|---|
bianass | ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bianass.1 | . . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | |
2 | 1 | anbi2i 454 | . 2 ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒))) |
3 | anass 399 | . 2 ⊢ (((𝜂 ∧ 𝜓) ∧ 𝜒) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒))) | |
4 | 2, 3 | bitr4i 186 | 1 ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: bianassc 467 |
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