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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.21t | Structured version Visualization version GIF version |
Description: Proof of 19.21t 2135 from stdpc5t 33641. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-19.21t | ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc5t 33641 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | |
2 | 19.9t 2133 | . . . 4 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 2 | imbi1d 334 | . . 3 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
4 | 19.38 1801 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
5 | 3, 4 | syl6bir 246 | . 2 ⊢ (Ⅎ𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))) |
6 | 1, 5 | impbid 204 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1505 ∃wex 1742 Ⅎwnf 1746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-12 2106 |
This theorem depends on definitions: df-bi 199 df-ex 1743 df-nf 1747 |
This theorem is referenced by: (None) |
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