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Mirrors > Home > MPE Home > Th. List > 19.42 | Structured version Visualization version GIF version |
Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 1950 for a version requiring fewer axioms. See exan 1858 for an immediate version. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
19.42.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.42 | ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | 19.41 2224 | . 2 ⊢ (∃𝑥(𝜓 ∧ 𝜑) ↔ (∃𝑥𝜓 ∧ 𝜑)) |
3 | exancom 1857 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | |
4 | ancom 460 | . 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓 ∧ 𝜑)) | |
5 | 2, 3, 4 | 3bitr4i 303 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∃wex 1774 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 |
This theorem is referenced by: eean 2340 bnj596 34377 bnj916 34564 bnj983 34582 |
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