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Mirrors > Home > ILE Home > Th. List > 19.33b2 | GIF version |
Description: The antecedent provides a condition implying the converse of 19.33 1472. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1618 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.) |
Ref | Expression |
---|---|
19.33b2 | ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orcom 718 | . . . . 5 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑)) | |
2 | alnex 1487 | . . . . . 6 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
3 | alnex 1487 | . . . . . 6 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
4 | 2, 3 | orbi12i 754 | . . . . 5 ⊢ ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑)) |
5 | 1, 4 | bitr4i 186 | . . . 4 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑)) |
6 | pm2.53 712 | . . . . . . 7 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑)) | |
7 | 6 | orcoms 720 | . . . . . 6 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑)) |
8 | 7 | al2imi 1446 | . . . . 5 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥 ¬ 𝜓 → ∀𝑥𝜑)) |
9 | pm2.53 712 | . . . . . 6 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | |
10 | 9 | al2imi 1446 | . . . . 5 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥 ¬ 𝜑 → ∀𝑥𝜓)) |
11 | 8, 10 | orim12d 776 | . . . 4 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥𝜑 ∨ ∀𝑥𝜓))) |
12 | 5, 11 | syl5bi 151 | . . 3 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓))) |
13 | 12 | com12 30 | . 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓))) |
14 | 19.33 1472 | . 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) | |
15 | 13, 14 | impbid1 141 | 1 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 698 ∀wal 1341 ∃wex 1480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-gen 1437 ax-ie2 1482 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-fal 1349 |
This theorem is referenced by: 19.33bdc 1618 |
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