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Mirrors > Home > MPE Home > Th. List > 19.21t | Structured version Visualization version GIF version |
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2172. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1766 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.) |
Ref | Expression |
---|---|
19.21t | ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.38a 1821 | . 2 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | |
2 | 19.9t 2169 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 2 | imbi1d 343 | . 2 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
4 | 1, 3 | bitr3d 282 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1520 ∃wex 1761 Ⅎwnf 1765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-12 2141 |
This theorem depends on definitions: df-bi 208 df-ex 1762 df-nf 1766 |
This theorem is referenced by: 19.21 2172 sbal1 2525 sbal2 2526 sbal2OLD 2527 sbal2OLDOLD 2528 r19.21t 3181 ceqsalt 3470 sbciegft 3740 bj-ceqsalt0 33783 bj-ceqsalt1 33784 wl-sbhbt 34347 wl-2sb6d 34351 wl-sbalnae 34355 wl-dfralf 34396 ax12indalem 35638 ax12inda2ALT 35639 |
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