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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 2203 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1979 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 2181 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | 19.3.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nf5ri 2193 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
4 | 1, 3 | impbii 209 | 1 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ 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: 19.16 2223 19.17 2224 19.27 2225 19.28 2226 19.37 2230 aaan 2332 axrep4 5291 axrep4OLD 5292 zfcndrep 10652 bj-alexbiex 36682 bj-alalbial 36684 fvineqsneq 37395 |
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