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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 41 | . . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | |
2 | 1 | aleximi 1749 | . . 3 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)) |
3 | 2 | com12 32 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
4 | exnal 1744 | . . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
5 | pm2.21 119 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
6 | 5 | eximi 1752 | . . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓)) |
7 | 4, 6 | sylbir 224 | . . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓)) |
8 | exa1 1756 | . . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓)) | |
9 | 7, 8 | ja 172 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
10 | 3, 9 | impbii 198 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: 19.35i 1795 19.35ri 1796 19.25 1797 19.43 1799 nfimd 1812 speimfwALT 1864 19.39 1886 19.24 1887 19.36v 1891 19.37v 1897 19.36 2085 19.37 2087 spimt 2241 grothprim 9535 bj-nalnaleximiOLD 31798 bj-spimt2 31896 bj-spimtv 31905 bj-nfimt 32025 bj-snsetex 32144 |
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