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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.42t | Structured version Visualization version GIF version |
Description: Closed form of 19.42 2233 from the same axioms as 19.42v 1961. (Contributed by BJ, 2-Dec-2023.) |
Ref | Expression |
---|---|
bj-19.42t | ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1893 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓)) | |
2 | bj-nnfe 34909 | . . . 4 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | |
3 | 2 | anim1d 611 | . . 3 ⊢ (Ⅎ'𝑥𝜑 → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (𝜑 ∧ ∃𝑥𝜓))) |
4 | 1, 3 | syl5 34 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑥𝜓))) |
5 | bj-nnfa 34906 | . . . 4 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
6 | 5 | anim1d 611 | . . 3 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → (∀𝑥𝜑 ∧ ∃𝑥𝜓))) |
7 | 19.29 1880 | . . 3 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | |
8 | 6, 7 | syl6 35 | . 2 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
9 | 4, 8 | impbid 211 | 1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1540 ∃wex 1786 Ⅎ'wnnf 34901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-bj-nnf 34902 |
This theorem is referenced by: bj-19.41t 34952 |
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