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Mirrors > Home > MPE Home > Th. List > 2ralor | Structured version Visualization version GIF version |
Description: Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Wolf Lammen, 20-Nov-2024.) |
Ref | Expression |
---|---|
2ralor | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.32v 3268 | . . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓)) | |
2 | orcom 867 | . . . 4 ⊢ ((𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑)) | |
3 | 1, 2 | bitri 274 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑)) |
4 | 3 | ralbii 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑)) |
5 | r19.32v 3268 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∨ 𝜑) ↔ (∀𝑦 ∈ 𝐵 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑)) | |
6 | orcom 867 | . 2 ⊢ ((∀𝑦 ∈ 𝐵 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓)) | |
7 | 4, 5, 6 | 3bitri 297 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 844 ∀wral 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1783 df-ral 3069 |
This theorem is referenced by: prmidl2 31602 ispridl2 36182 |
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