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| Mirrors > Home > MPE Home > Th. List > 19.3 | Structured version Visualization version GIF version | ||
| Description: A wff may be quantified with a variable not free in it. Version of 19.9 2205 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1981 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
| Ref | Expression |
|---|---|
| 19.3.1 | ⊢ Ⅎ𝑥𝜑 |
| Ref | Expression |
|---|---|
| 19.3 | ⊢ (∀𝑥𝜑 ↔ 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sp 2183 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
| 2 | 19.3.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 3 | 2 | nf5ri 2195 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
| 4 | 1, 3 | impbii 209 | 1 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: 19.16 2225 19.17 2226 19.27 2227 19.28 2228 19.37 2232 aaan 2333 axrep4 5285 axrep4OLD 5286 zfcndrep 10654 bj-alexbiex 36700 bj-alalbial 36702 fvineqsneq 37413 |
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