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Mirrors > Home > MPE Home > Th. List > 19.9d | Structured version Visualization version GIF version |
Description: A deduction version of one direction of 19.9 2072. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
Ref | Expression |
---|---|
19.9d.1 | ⊢ (𝜓 → Ⅎ𝑥𝜑) |
Ref | Expression |
---|---|
19.9d | ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9d.1 | . . 3 ⊢ (𝜓 → Ⅎ𝑥𝜑) | |
2 | df-nf 1710 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | sylib 208 | . 2 ⊢ (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | sp 2053 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
5 | 3, 4 | syl6 35 | 1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-ex 1705 df-nf 1710 |
This theorem is referenced by: 19.9t 2071 exdistrf 2333 equvel 2347 copsexg 4956 19.9d2rf 29318 wl-exeq 33321 spd 42425 |
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