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Mirrors > Home > MPE Home > Th. List > 19.9v | Structured version Visualization version GIF version |
Description: Version of 19.9 2199 with a disjoint variable condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1986. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2012. (Revised by Wolf Lammen, 4-Dec-2017.) |
Ref | Expression |
---|---|
19.9v | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax5e 1916 | . 2 ⊢ (∃𝑥𝜑 → 𝜑) | |
2 | 19.8v 1987 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
3 | 1, 2 | impbii 208 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1782 |
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 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: 19.36v 1992 19.44v 1997 19.45v 1998 zfcndpow 10611 volfiniune 33228 bnj937 33782 bnj594 33923 bnj907 33978 bnj1128 34001 bnj1145 34004 coss0 37349 prter2 37751 relopabVD 43662 rfcnnnub 43720 |
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