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| Mirrors > Home > MPE Home > Th. List > r19.32v | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of 19.32v 1939. (Contributed by NM, 25-Nov-2003.) | 
| Ref | Expression | 
|---|---|
| r19.32v | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | r19.21v 3179 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | |
| 2 | df-or 848 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
| 3 | 2 | ralbii 3092 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓)) | 
| 4 | df-or 848 | . 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | |
| 5 | 1, 3, 4 | 3bitr4i 303 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 ∀wral 3060 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ral 3061 | 
| This theorem is referenced by: 2ralor 3230 iinun2 5072 iinuni 5097 axcontlem2 28981 axcontlem7 28986 disjnf 32584 lindslinindsimp2 48385 | 
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