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| Mirrors > Home > MPE Home > Th. List > 19.32 | Structured version Visualization version GIF version | ||
| Description: Theorem 19.32 of [Margaris] p. 90. See 19.32v 1940 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
| Ref | Expression |
|---|---|
| 19.32.1 | ⊢ Ⅎ𝑥𝜑 |
| Ref | Expression |
|---|---|
| 19.32 | ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.32.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 1 | nfn 1857 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
| 3 | 2 | 19.21 2207 | . 2 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)) |
| 4 | df-or 849 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
| 5 | 4 | albii 1819 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓)) |
| 6 | df-or 849 | . 2 ⊢ ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)) | |
| 7 | 3, 5, 6 | 3bitr4i 303 | 1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 848 ∀wal 1538 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: 19.31 2234 2eu3 2654 axi12 2706 axbnd 2707 |
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