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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 1809. (Contributed by Scott Fenton, 25-Feb-2011.) |
Ref | Expression |
---|---|
r19.30 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralim 2948 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)) | |
2 | orcom 402 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
3 | df-or 385 | . . . 4 ⊢ ((𝜓 ∨ 𝜑) ↔ (¬ 𝜓 → 𝜑)) | |
4 | 2, 3 | bitri 264 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜓 → 𝜑)) |
5 | 4 | ralbii 2980 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑)) |
6 | orcom 402 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑)) | |
7 | dfrex2 2996 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓) | |
8 | 7 | orbi2i 541 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓)) |
9 | imor 428 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑)) | |
10 | 6, 8, 9 | 3bitr4i 292 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)) |
11 | 1, 5, 10 | 3imtr4i 281 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∀wral 2912 ∃wrex 2913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-ral 2917 df-rex 2918 |
This theorem is referenced by: disjunsn 29407 esumcvg 30148 |
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