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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 1907 | . . 3 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)) |
3 | 2 | com12 32 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
4 | exnal 1902 | . . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
5 | pm2.21 121 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
6 | 5 | eximi 1910 | . . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓)) |
7 | 4, 6 | sylbir 225 | . . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓)) |
8 | exa1 1913 | . . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓)) | |
9 | 7, 8 | ja 174 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
10 | 3, 9 | impbii 199 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1629 ∃wex 1852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 |
This theorem depends on definitions: df-bi 197 df-ex 1853 |
This theorem is referenced by: 19.35i 1958 19.35ri 1959 19.25 1960 19.43 1962 nfimd 1973 nfimdOLDOLD 1975 19.36imv 2025 19.37imv 2027 speimfwALT 2046 19.39 2068 19.24 2069 19.36v 2072 19.37v 2078 19.36 2254 19.37 2256 spimt 2415 grothprim 9856 bj-nfimt 32947 bj-spimt2 33039 bj-spimtv 33048 bj-snsetex 33275 |
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