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Mirrors > Home > MPE Home > Th. List > 19.38 | Structured version Visualization version GIF version |
Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see 19.38a 1835 and 19.38b 1836. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2195. (Revised by Wolf Lammen, 2-Jan-2018.) |
Ref | Expression |
---|---|
19.38 | ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1776 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
2 | pm2.21 123 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
3 | 2 | alimi 1806 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓)) |
4 | 1, 3 | sylbir 234 | . 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓)) |
5 | ala1 1808 | . 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓)) | |
6 | 4, 5 | ja 186 | 1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1532 ∃wex 1774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 |
This theorem depends on definitions: df-bi 206 df-ex 1775 |
This theorem is referenced by: 19.38a 1835 19.38b 1836 nfimd 1890 19.21v 1935 19.23v 1938 bj-nfimexal 36102 bj-nfimt 36114 bj-wnf1 36194 bj-substax12 36198 bj-nnfim 36223 bj-19.21t 36246 bj-19.23t 36247 bj-19.21t0 36307 pm10.53 43803 |
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