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Mirrors > Home > MPE Home > Th. List > 19.41 | Structured version Visualization version GIF version |
Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1941 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
Ref | Expression |
---|---|
19.41.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.41 | ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1878 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓)) | |
2 | 19.41.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 2195 | . . . 4 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 3 | anbi2i 622 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
5 | 1, 4 | sylib 219 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ 𝜓)) |
6 | pm3.21 472 | . . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
7 | 2, 6 | eximd 2206 | . . 3 ⊢ (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
8 | 7 | impcom 408 | . 2 ⊢ ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
9 | 5, 8 | impbii 210 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: 19.42 2228 equsexv 2259 eean 2360 eeeanv 2362 equsexALT 2432 2sb5rf 2488 r19.41 3345 eliunxp 5701 dfopab2 7739 dfoprab3s 7740 xpcomco 8595 mpomptxf 30353 bnj605 32078 bnj607 32087 2sb5nd 40771 2sb5ndVD 41121 2sb5ndALT 41143 eliunxp2 44310 |
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