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Mirrors > Home > MPE Home > Th. List > 19.36v | Structured version Visualization version GIF version |
Description: Version of 19.36 2162 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 1840 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.9v 1938 | . . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓) | |
3 | 2 | imbi2i 328 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | bitri 267 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1505 ∃wex 1742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 |
This theorem depends on definitions: df-bi 199 df-ex 1743 |
This theorem is referenced by: 19.12vvv 1945 19.12vv 2281 ax13lem2 2305 axext2 2745 vtocl2OLD 3475 spcimdv 3505 bnj1090 31925 bj-spimvwt 33541 bj-spcimdv 33733 bj-spcimdvv 33734 19.36vv 40160 |
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