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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralbidb | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 45726 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.) |
Ref | Expression |
---|---|
ralbidb.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) |
ralbidb.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
ralbidb | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbidb.1 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) | |
2 | ralbidb.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃)) | |
3 | 1, 2 | logic1a 45718 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜒) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜓) → 𝜃))) |
4 | impexp 454 | . . 3 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) → 𝜃) ↔ (𝑥 ∈ 𝐵 → (𝜓 → 𝜃))) | |
5 | 3, 4 | bitrdi 290 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜒) ↔ (𝑥 ∈ 𝐵 → (𝜓 → 𝜃)))) |
6 | 5 | ralbidv2 3108 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ∀wral 3054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ral 3059 |
This theorem is referenced by: (None) |
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