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Mirrors > Home > ILE Home > Th. List > 3anassrs | GIF version |
Description: Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
3anassrs.1 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) |
Ref | Expression |
---|---|
3anassrs | ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anassrs.1 | . . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) | |
2 | 1 | 3exp2 1225 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
3 | 2 | imp41 353 | 1 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 |
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 df-3an 980 |
This theorem is referenced by: ralrimivvva 2560 euotd 4254 dfgrp3me 12924 neitx 13661 xmetpsmet 13762 |
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