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| Mirrors > Home > MPE Home > Th. List > 19.23 | Structured version Visualization version GIF version | ||
| Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1942 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
| Ref | Expression |
|---|---|
| 19.23.1 | ⊢ Ⅎ𝑥𝜓 |
| Ref | Expression |
|---|---|
| 19.23 | ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.23.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 2 | 19.23t 2210 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 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: exlimi 2217 equsalv 2267 nf5 2282 19.23h 2288 pm11.53 2348 equsal 2422 2sb6rf 2478 ceqsal 3519 r19.3rz 4497 ssrelf 32627 bj-biexal1 36706 bj-biexex 36710 axc11n-16 38939 axc11next 44425 r19.3rzf 45163 |
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