![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 19.36v | Structured version Visualization version GIF version |
Description: Version of 19.36 2219 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.) |
Ref | Expression |
---|---|
19.36v | ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1873 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.9v 1980 | . . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓) | |
3 | 2 | imbi2i 336 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | bitri 275 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 ∃wex 1774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 |
This theorem depends on definitions: df-bi 206 df-ex 1775 |
This theorem is referenced by: 19.12vvv 1985 19.12vv 2339 ax13lem2 2371 axexte 2700 spcimdv 3580 bnj1090 34610 bj-spimvwt 36145 bj-spcimdv 36373 bj-spcimdvv 36374 19.36vv 43820 |
Copyright terms: Public domain | W3C validator |