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Mirrors > Home > ILE Home > Th. List > pm3.2an3 | GIF version |
Description: pm3.2 138 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.) |
Ref | Expression |
---|---|
pm3.2an3 | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 ∧ 𝜓 ∧ 𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2 138 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → ((𝜑 ∧ 𝜓) ∧ 𝜒))) | |
2 | 1 | ex 114 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → ((𝜑 ∧ 𝜓) ∧ 𝜒)))) |
3 | df-3an 965 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
4 | 3 | bicomi 131 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
5 | 2, 4 | syl8ib 165 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 ∧ 𝜓 ∧ 𝜒)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 |
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 df-3an 965 |
This theorem is referenced by: 3exp 1184 |
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