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Mirrors > Home > MPE Home > Th. List > 19.36v | Structured version Visualization version GIF version |
Description: Version of 19.36 2228 with a disjoint variable condition instead of a non-freeness 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 1874 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.9v 1984 | . . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓) | |
3 | 2 | imbi2i 338 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | bitri 277 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 |
This theorem depends on definitions: df-bi 209 df-ex 1777 |
This theorem is referenced by: 19.12vvv 1991 19.12vv 2364 ax13lem2 2390 axexte 2794 vtocl2OLD 3562 spcimdv 3591 bnj1090 32251 bj-spimvwt 34002 bj-spcimdv 34211 bj-spcimdvv 34212 19.36vv 40713 |
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