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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 36117, 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 36117 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 36118 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 1881 | . . 3 ⊢ (∃𝑥(𝜓 ∧ 𝜒) → (∃𝑥𝜓 ∧ ∃𝑥𝜒)) | |
2 | bj-nnfand.1 | . . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜓) | |
3 | 2 | bj-nnfed 36101 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓)) |
4 | bj-nnfand.2 | . . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜒) | |
5 | 4 | bj-nnfed 36101 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒)) |
6 | 3, 5 | anim12d 608 | . . 3 ⊢ (𝜑 → ((∃𝑥𝜓 ∧ ∃𝑥𝜒) → (𝜓 ∧ 𝜒))) |
7 | 1, 6 | syl5 34 | . 2 ⊢ (𝜑 → (∃𝑥(𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))) |
8 | 2 | bj-nnfad 36098 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
9 | 4 | bj-nnfad 36098 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
10 | 8, 9 | anim12d 608 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜒))) |
11 | 19.26 1865 | . . 3 ⊢ (∀𝑥(𝜓 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒)) | |
12 | 10, 11 | imbitrrdi 251 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒))) |
13 | df-bj-nnf 36093 | . 2 ⊢ (Ⅎ'𝑥(𝜓 ∧ 𝜒) ↔ ((∃𝑥(𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)) ∧ ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒)))) | |
14 | 7, 12, 13 | sylanbrc 582 | 1 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 ∃wex 1773 Ⅎ'wnnf 36092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-bj-nnf 36093 |
This theorem is referenced by: bj-nnfbid 36122 |
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