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Mirrors > Home > ILE Home > Th. List > 19.32r | GIF version |
Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1569. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Ref | Expression |
---|---|
19.32r.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.32r | ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.32r.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | orc 633 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 1, 2 | alrimi 1415 | . 2 ⊢ (𝜑 → ∀𝑥(𝜑 ∨ 𝜓)) |
4 | olc 632 | . . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
5 | 4 | alimi 1344 | . 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 ∨ 𝜓)) |
6 | 3, 5 | jaoi 636 | 1 ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 ∀wal 1241 Ⅎwnf 1349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-gen 1338 ax-4 1400 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: 19.31r 1571 |
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