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Mirrors > Home > MPE Home > Th. List > r19.32v | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.32v 1941. (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
r19.32v | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.21v 3177 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | |
2 | df-or 844 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
3 | 2 | ralbii 3167 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓)) |
4 | df-or 844 | . 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | |
5 | 1, 3, 4 | 3bitr4i 305 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 ∀wral 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1781 df-ral 3145 |
This theorem is referenced by: iinun2 4997 iinuni 5022 axcontlem2 26753 axcontlem7 26758 disjnf 30322 lindslinindsimp2 44525 |
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