![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 1924 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 2177 | . 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 207 ∀wal 1523 ∃wex 1765 Ⅎwnf 1769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-12 2143 |
This theorem depends on definitions: df-bi 208 df-ex 1766 df-nf 1770 |
This theorem is referenced by: exlimi 2184 equsalv 2233 nf5 2258 19.23h 2264 pm11.53 2325 equsal 2397 2sb6rf 2456 2sb6rfOLD 2457 r19.3rz 4362 ralidm 4375 ssrelf 30052 bj-biexal1 33643 bj-biexex 33647 axc11n-16 35626 axc11next 40297 |
Copyright terms: Public domain | W3C validator |