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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfand | Structured version Visualization version GIF version |
Description: Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 36731, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 36731 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 36732 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnfand.1 | ⊢ (𝜑 → Ⅎ'𝑥𝜓) |
bj-nnfand.2 | ⊢ (𝜑 → Ⅎ'𝑥𝜒) |
Ref | Expression |
---|---|
bj-nnfand | ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1884 | . . 3 ⊢ (∃𝑥(𝜓 ∧ 𝜒) → (∃𝑥𝜓 ∧ ∃𝑥𝜒)) | |
2 | bj-nnfand.1 | . . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜓) | |
3 | 2 | bj-nnfed 36715 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓)) |
4 | bj-nnfand.2 | . . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜒) | |
5 | 4 | bj-nnfed 36715 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒)) |
6 | 3, 5 | anim12d 609 | . . 3 ⊢ (𝜑 → ((∃𝑥𝜓 ∧ ∃𝑥𝜒) → (𝜓 ∧ 𝜒))) |
7 | 1, 6 | syl5 34 | . 2 ⊢ (𝜑 → (∃𝑥(𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))) |
8 | 2 | bj-nnfad 36712 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
9 | 4 | bj-nnfad 36712 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
10 | 8, 9 | anim12d 609 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜒))) |
11 | 19.26 1868 | . . 3 ⊢ (∀𝑥(𝜓 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒)) | |
12 | 10, 11 | imbitrrdi 252 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒))) |
13 | df-bj-nnf 36707 | . 2 ⊢ (Ⅎ'𝑥(𝜓 ∧ 𝜒) ↔ ((∃𝑥(𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)) ∧ ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒)))) | |
14 | 7, 12, 13 | sylanbrc 583 | 1 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∃wex 1776 Ⅎ'wnnf 36706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-bj-nnf 36707 |
This theorem is referenced by: bj-nnfbid 36736 |
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