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Mirrors > Home > MPE Home > Th. List > 19.35 | Structured version Visualization version GIF version |
Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |
Ref | Expression |
---|---|
19.35 | ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.27 42 | . . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | |
2 | 1 | aleximi 1829 | . . 3 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)) |
3 | 2 | com12 32 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
4 | exnal 1824 | . . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
5 | pm2.21 123 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
6 | 5 | eximi 1832 | . . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓)) |
7 | 4, 6 | sylbir 235 | . . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓)) |
8 | exa1 1835 | . . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓)) | |
9 | 7, 8 | ja 186 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
10 | 3, 9 | impbii 209 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1535 ∃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 |
This theorem depends on definitions: df-bi 207 df-ex 1777 |
This theorem is referenced by: 19.35i 1876 19.35ri 1877 19.25 1878 19.43 1880 nfimd 1892 19.37imv 1945 speimfwALT 1962 19.39 1982 19.24 1983 19.36v 1985 19.37v 1989 19.36 2228 19.37 2230 spimt 2389 grothprim 10872 bj-nfimt 36621 bj-nnfim 36729 bj-19.36im 36754 bj-19.37im 36755 bj-spimt2 36768 bj-spimtv 36777 |
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