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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 1932 for a version requiring fewer axioms. See exan 1844 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 2201 | . 2 ⊢ (∃𝑥(𝜓 ∧ 𝜑) ↔ (∃𝑥𝜓 ∧ 𝜑)) |
3 | exancom 1843 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | |
4 | ancom 461 | . 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓 ∧ 𝜑)) | |
5 | 2, 3, 4 | 3bitr4i 304 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1762 Ⅎwnf 1766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-12 2140 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1763 df-nf 1767 |
This theorem is referenced by: eean 2324 bnj596 31626 bnj916 31813 bnj983 31831 |
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