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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 2203. (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 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 8-Jul-2022.) |
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 | 1 | nfrd 1793 | . 2 ⊢ (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
3 | sp 2180 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
4 | 2, 3 | syl6 35 | 1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∃wex 1781 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1782 df-nf 1786 |
This theorem is referenced by: 19.9t 2202 19.9ht 2328 spimt 2393 exdistrf 2458 equvel 2468 copsexgw 5346 copsexg 5347 oprabidw 7166 19.9d2rf 30242 copsex2d 34554 wl-exeq 34939 spd 45208 |
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