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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 2096 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 1980 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.37.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | 19.3 2243 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
4 | 3 | imbi1i 341 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
5 | 1, 4 | bitri 267 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 ∃wex 1878 Ⅎwnf 1882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-ex 1879 df-nf 1883 |
This theorem is referenced by: bnj900 31541 |
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