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Mirrors > Home > MPE Home > Th. List > 19.23t | Structured version Visualization version GIF version |
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2209. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 8-Oct-2022.) |
Ref | Expression |
---|---|
19.23t | ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.38b 1842 | . 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | |
2 | 19.3t 2199 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥𝜓 ↔ 𝜓)) | |
3 | 2 | imbi2d 344 | . 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
4 | 1, 3 | bitr3d 284 | 1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀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: 19.23 2209 axie2 2765 r19.23t 3272 ceqsalt 3474 vtoclgft 3501 vtoclgftOLD 3502 sbciegft 3756 bj-ceqsalt0 34324 bj-ceqsalt1 34325 wl-equsald 34944 |
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