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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm11.62 | Structured version Visualization version GIF version |
Description: Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm11.62 | ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impexp 461 | . . . 4 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) | |
2 | 1 | albii 1737 | . . 3 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑦(𝜑 → (𝜓 → 𝜒))) |
3 | 19.21v 1855 | . . 3 ⊢ (∀𝑦(𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → ∀𝑦(𝜓 → 𝜒))) | |
4 | 2, 3 | bitri 263 | . 2 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → ∀𝑦(𝜓 → 𝜒))) |
5 | 4 | albii 1737 | 1 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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