|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > r19.30OLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of 19.30 1881 as of 5-Nov-2024. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| r19.30OLD | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pm2.53 852 | . . . 4 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑)) | |
| 2 | 1 | orcoms 873 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑)) | 
| 3 | 2 | ralimi 3083 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → ∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑)) | 
| 4 | ralim 3086 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)) | |
| 5 | ralnex 3072 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
| 6 | 5 | biimpri 228 | . . . . 5 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜓) | 
| 7 | 6 | imim1i 63 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) → (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)) | 
| 8 | 7 | orrd 864 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑)) | 
| 9 | 8 | orcomd 872 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | 
| 10 | 3, 4, 9 | 3syl 18 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∨ wo 848 ∀wral 3061 ∃wrex 3070 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-ral 3062 df-rex 3071 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |