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Mirrors > Home > MPE Home > Th. List > r19.30 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.30 1882. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.) |
Ref | Expression |
---|---|
r19.30 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.53 847 | . . . 4 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑)) | |
2 | 1 | orcoms 868 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑)) |
3 | 2 | ralimi 3162 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → ∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑)) |
4 | ralim 3164 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)) | |
5 | ralnex 3238 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
6 | 5 | biimpri 230 | . . . . 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 859 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑)) |
9 | 8 | orcomd 867 | . 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 843 ∀wral 3140 ∃wrex 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-ral 3145 df-rex 3146 |
This theorem is referenced by: disjunsn 30346 esumcvg 31347 |
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