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Mirrors > Home > MPE Home > Th. List > 19.9v | Structured version Visualization version GIF version |
Description: Version of 19.9 2227 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 2065. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2092. (Revised by Wolf Lammen, 4-Dec-2017.) |
Ref | Expression |
---|---|
19.9v | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax5e 1992 | . 2 ⊢ (∃𝑥𝜑 → 𝜑) | |
2 | 19.8v 2063 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
3 | 1, 2 | impbii 199 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∃wex 1851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 |
This theorem depends on definitions: df-bi 197 df-ex 1852 |
This theorem is referenced by: 19.3v 2065 19.23vOLD 2070 19.36v 2071 19.44v 2079 19.45v 2080 19.41vOLD 2081 zfcndpow 9639 volfiniune 30627 bnj937 31174 bnj594 31314 bnj907 31367 bnj1128 31390 bnj1145 31393 bj-sbfvv 33095 coss0 34564 prter2 34682 relopabVD 39653 rfcnnnub 39711 |
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