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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 1943 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 208 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-nf 1785 |
This theorem is referenced by: exlimi 2217 equsalv 2268 nf5 2290 19.23h 2296 pm11.53 2367 equsal 2439 2sb6rf 2497 r19.3rz 4442 ralidm 4455 ssrelf 30366 bj-biexal1 34039 bj-biexex 34043 axc11n-16 36089 axc11next 40758 |
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