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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 2014 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 1929 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
3 | 2 | 19.21 2218 | . 2 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)) |
4 | df-or 384 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
5 | 4 | albii 1892 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓)) |
6 | df-or 384 | . 2 ⊢ ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)) | |
7 | 3, 5, 6 | 3bitr4i 292 | 1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∀wal 1626 Ⅎwnf 1853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-12 2192 |
This theorem depends on definitions: df-bi 197 df-or 384 df-ex 1850 df-nf 1855 |
This theorem is referenced by: 19.31 2245 2eu3 2689 axi12 2734 |
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