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Mirrors > Home > MPE Home > Th. List > 19.21 | Structured version Visualization version GIF version |
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1937 for a version requiring fewer axioms. See also 19.21h 2286. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1781 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
Ref | Expression |
---|---|
19.21.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.21 | ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 19.21t 2204 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-ex 1777 df-nf 1781 |
This theorem is referenced by: stdpc5 2206 19.21-2 2207 19.32 2231 nf6 2282 19.21h 2286 sbrim 2303 cbv1v 2337 19.12vv 2348 cbv1 2405 axc14 2466 r2alf 3279 19.12b 35783 bj-biexal2 36689 bj-bialal 36691 wl-dral1d 37512 mpobi123f 38149 |
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