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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.37im | Structured version Visualization version GIF version |
Description: One direction of 19.37 2217 from the same axioms as 19.37imv 1943. (Contributed by BJ, 2-Dec-2023.) |
Ref | Expression |
---|---|
bj-19.37im | ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1872 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | bj-nnfa 36096 | . . 3 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
3 | 2 | imim1d 82 | . 2 ⊢ (Ⅎ'𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → ∃𝑥𝜓))) |
4 | 1, 3 | biimtrid 241 | 1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1773 Ⅎ'wnnf 36091 |
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 36092 |
This theorem is referenced by: (None) |
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