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Mirrors > Home > MPE Home > Th. List > r2allem | Structured version Visualization version GIF version |
Description: Lemma factoring out common proof steps of r2alf 2938 and r2al 2939. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 9-Jan-2020.) |
Ref | Expression |
---|---|
r2allem.1 | ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) |
Ref | Expression |
---|---|
r2allem | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2917 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)) | |
2 | r2allem.1 | . . . 4 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) | |
3 | impexp 462 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑))) | |
4 | 3 | albii 1747 | . . . 4 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑))) |
5 | df-ral 2917 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)) | |
6 | 5 | imbi2i 326 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) |
7 | 2, 4, 6 | 3bitr4i 292 | . . 3 ⊢ (∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)) |
8 | 7 | albii 1747 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)) |
9 | 1, 8 | bitr4i 267 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ral 2917 |
This theorem is referenced by: r2alf 2938 r2al 2939 |
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