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Mirrors > Home > ILE Home > Th. List > pm4.71d | GIF version |
Description: Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
pm4.71rd.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
pm4.71d | ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.71rd.1 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | pm4.71 389 | . 2 ⊢ ((𝜓 → 𝜒) ↔ (𝜓 ↔ (𝜓 ∧ 𝜒))) | |
3 | 1, 2 | sylib 122 | 1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: difin2 3397 resopab2 4954 fcnvres 5399 resoprab2 5971 cndis 13672 cnpdis 13673 blpnf 13831 |
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