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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 1856 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 1768 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
3 | 2 | 19.21 2062 | . 2 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)) |
4 | df-or 384 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
5 | 4 | albii 1737 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓)) |
6 | df-or 384 | . 2 ⊢ ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)) | |
7 | 3, 5, 6 | 3bitr4i 291 | 1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: 19.31 2089 2eu3 2543 axi12 2588 |
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