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Mirrors > Home > MPE Home > Th. List > 19.36 | Structured version Visualization version GIF version |
Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1999 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.) |
Ref | Expression |
---|---|
19.36.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.36 | ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1884 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.36.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 2207 | . . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 3 | imbi2i 339 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
5 | 1, 4 | bitri 278 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-ex 1787 df-nf 1791 |
This theorem is referenced by: 19.36i 2233 19.12vv 2350 spcimgft 3491 19.12b 33351 |
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