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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 1938 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 1783 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 2205 | . 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 1537 Ⅎwnf 1782 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 df-nf 1783 | 
| This theorem is referenced by: stdpc5 2207 19.21-2 2208 19.32 2232 nf6 2282 19.21h 2286 sbrim 2303 cbv1v 2337 19.12vv 2348 cbv1 2406 axc14 2467 r2alf 3280 19.12b 35803 bj-biexal2 36708 bj-bialal 36710 wl-dral1d 37533 mpobi123f 38170 | 
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