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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 1954 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 1890 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓)) | |
2 | 19.41.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 2199 | . . . 4 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 3 | anbi2i 624 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
5 | 1, 4 | sylib 217 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ 𝜓)) |
6 | pm3.21 473 | . . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
7 | 2, 6 | eximd 2210 | . . 3 ⊢ (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
8 | 7 | impcom 409 | . 2 ⊢ ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
9 | 5, 8 | impbii 208 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 |
This theorem is referenced by: 19.42 2230 eean 2345 eeeanv 2347 equsexALT 2419 2sb5rf 2472 r19.41 3261 eliunxp 5838 dfopab2 8038 dfoprab3s 8039 xpcomco 9062 mpomptxf 31905 bnj605 33918 bnj607 33927 2sb5nd 43321 2sb5ndVD 43671 2sb5ndALT 43693 eliunxp2 47009 |
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