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| Mirrors > Home > MPE Home > Th. List > 19.9v | Structured version Visualization version GIF version | ||
| Description: Version of 19.9 2205 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 1981. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2007. (Revised by Wolf Lammen, 4-Dec-2017.) | 
| Ref | Expression | 
|---|---|
| 19.9v | ⊢ (∃𝑥𝜑 ↔ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax5e 1912 | . 2 ⊢ (∃𝑥𝜑 → 𝜑) | |
| 2 | 19.8v 1982 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
| 3 | 1, 2 | impbii 209 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∃wex 1779 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 | 
| This theorem is referenced by: 19.36v 1987 19.44v 1992 19.45v 1993 zfcndpow 10656 volfiniune 34231 bnj937 34785 bnj594 34926 bnj907 34981 bnj1128 35004 bnj1145 35007 coss0 38480 prter2 38882 relopabVD 44921 rfcnnnub 45041 | 
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