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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4an4132 | Structured version Visualization version GIF version |
Description: A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.) |
Ref | Expression |
---|---|
4an4132.1 | ⊢ ((((𝜃 ∧ 𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏) |
Ref | Expression |
---|---|
4an4132 | ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜃) | |
2 | simplr 769 | . . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜒) | |
3 | 1, 2 | jca 515 | . 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → (𝜃 ∧ 𝜒)) |
4 | simpllr 776 | . 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓) | |
5 | simplll 775 | . 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑) | |
6 | 4an4132.1 | . 2 ⊢ ((((𝜃 ∧ 𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏) | |
7 | 3, 4, 5, 6 | syl21anc 837 | 1 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: sineq0ALT 42136 |
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