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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 3254. (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 1865 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
3 | 2 | r19.21 3136 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
4 | df-or 848 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
5 | 4 | ralbii 3088 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓)) |
6 | df-or 848 | . 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | |
7 | 3, 5, 6 | 3bitr4i 306 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 847 Ⅎwnf 1791 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-or 848 df-ex 1788 df-nf 1792 df-ral 3066 |
This theorem is referenced by: 2reu3 44274 |
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