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| Mirrors > Home > MPE Home > Th. List > ralin | Structured version Visualization version GIF version | ||
| Description: Restricted universal quantification over intersection. (Contributed by Peter Mazsa, 8-Sep-2023.) |
| Ref | Expression |
|---|---|
| ralin | ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elin 3967 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
| 2 | 1 | imbi1i 349 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑)) |
| 3 | impexp 450 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑))) | |
| 4 | 2, 3 | bitri 275 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑))) |
| 5 | 4 | ralbii2 3089 | 1 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-in 3958 |
| This theorem is referenced by: ref5 38314 sswfaxreg 45004 |
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