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Mirrors > Home > MPE Home > Th. List > 19.23t | Structured version Visualization version GIF version |
Description: Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2067. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
Ref | Expression |
---|---|
19.23t | ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnt 1767 | . . 3 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥 ¬ 𝜓) | |
2 | 19.21t 2061 | . . 3 ⊢ (Ⅎ𝑥 ¬ 𝜓 → (∀𝑥(¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))) |
4 | con34b 305 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)) | |
5 | 4 | albii 1737 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(¬ 𝜓 → ¬ 𝜑)) |
6 | eximal 1698 | . 2 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)) | |
7 | 3, 5, 6 | 3bitr4g 302 | 1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: 19.23 2067 axie2 2585 r19.23t 3003 ceqsalt 3201 vtoclgft 3227 vtoclgftOLD 3228 sbciegft 3433 bj-ceqsalt0 32067 bj-ceqsalt1 32068 wl-equsald 32504 |
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