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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.21t | Structured version Visualization version GIF version |
Description: Statement 19.21t 2204 proved from modalK (obsoleting 19.21v 1947). (Contributed by BJ, 2-Dec-2023.) |
Ref | Expression |
---|---|
bj-19.21t | ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnf-alrim 34674 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | |
2 | bj-nnfe 34650 | . . . 4 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | |
3 | 2 | imim1d 82 | . . 3 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
4 | 19.38 1846 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
5 | 3, 4 | syl6 35 | . 2 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))) |
6 | 1, 5 | impbid 215 | 1 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀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-ex 1788 df-bj-nnf 34643 |
This theorem is referenced by: bj-pm11.53vw 34695 |
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