![]() |
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 1946 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 2204 | . 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 205 ∀wal 1540 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-nf 1787 |
This theorem is referenced by: exlimi 2211 equsalv 2259 nf5 2279 19.23h 2285 pm11.53 2343 equsal 2416 2sb6rf 2472 ceqsal 3482 r19.3rz 4459 ralidmOLD 4478 ssrelf 31576 bj-biexal1 35199 bj-biexex 35203 axc11n-16 37429 axc11next 42760 r19.3rzf 43447 |
Copyright terms: Public domain | W3C validator |