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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnford | Structured version Visualization version GIF version |
Description: Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 34669 and bj-nnfand 34668. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnford.1 | ⊢ (𝜑 → Ⅎ'𝑥𝜓) |
bj-nnford.2 | ⊢ (𝜑 → Ⅎ'𝑥𝜒) |
Ref | Expression |
---|---|
bj-nnford | ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.43 1890 | . . 3 ⊢ (∃𝑥(𝜓 ∨ 𝜒) ↔ (∃𝑥𝜓 ∨ ∃𝑥𝜒)) | |
2 | bj-nnford.1 | . . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜓) | |
3 | 2 | bj-nnfed 34651 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓)) |
4 | bj-nnford.2 | . . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜒) | |
5 | 4 | bj-nnfed 34651 | . . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒)) |
6 | 3, 5 | orim12d 965 | . . 3 ⊢ (𝜑 → ((∃𝑥𝜓 ∨ ∃𝑥𝜒) → (𝜓 ∨ 𝜒))) |
7 | 1, 6 | syl5bi 245 | . 2 ⊢ (𝜑 → (∃𝑥(𝜓 ∨ 𝜒) → (𝜓 ∨ 𝜒))) |
8 | 2 | bj-nnfad 34648 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
9 | 4 | bj-nnfad 34648 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
10 | 8, 9 | orim12d 965 | . . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) → (∀𝑥𝜓 ∨ ∀𝑥𝜒))) |
11 | 19.33 1892 | . . 3 ⊢ ((∀𝑥𝜓 ∨ ∀𝑥𝜒) → ∀𝑥(𝜓 ∨ 𝜒)) | |
12 | 10, 11 | syl6 35 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) → ∀𝑥(𝜓 ∨ 𝜒))) |
13 | df-bj-nnf 34643 | . 2 ⊢ (Ⅎ'𝑥(𝜓 ∨ 𝜒) ↔ ((∃𝑥(𝜓 ∨ 𝜒) → (𝜓 ∨ 𝜒)) ∧ ((𝜓 ∨ 𝜒) → ∀𝑥(𝜓 ∨ 𝜒)))) | |
14 | 7, 12, 13 | sylanbrc 586 | 1 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 ∀wal 1541 ∃wex 1787 Ⅎ'wnnf 34642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-bj-nnf 34643 |
This theorem is referenced by: (None) |
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