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Mirrors > Home > MPE Home > Th. List > 19.37 | Structured version Visualization version GIF version |
Description: Theorem 19.37 of [Margaris] p. 90. See 19.37v 2002 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
19.37.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.37 | ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1883 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.37.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | 19.3 2203 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
4 | 3 | imbi1i 353 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
5 | 1, 4 | bitri 278 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
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 1916 ax-6 1974 ax-7 2019 ax-12 2178 |
This theorem depends on definitions: df-bi 210 df-ex 1787 df-nf 1791 |
This theorem is referenced by: bnj900 32472 |
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