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Mirrors > Home > MPE Home > Th. List > Mathboxes > r19.32 | Structured version Visualization version GIF version |
Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3337. (Contributed by Alexander van der Vekens, 29-Jun-2017.) |
Ref | Expression |
---|---|
r19.32.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
r19.32 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.32.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfn 1848 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
3 | 2 | r19.21 3212 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
4 | df-or 842 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
5 | 4 | ralbii 3162 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓)) |
6 | df-or 842 | . 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | |
7 | 3, 5, 6 | 3bitr4i 304 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∨ wo 841 Ⅎwnf 1775 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-or 842 df-ex 1772 df-nf 1776 df-ral 3140 |
This theorem is referenced by: 2reu3 43186 |
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