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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 1986. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2015. (Revised by Wolf Lammen, 4-Dec-2017.) |
Ref | Expression |
---|---|
19.9v | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax5e 1913 | . 2 ⊢ (∃𝑥𝜑 → 𝜑) | |
2 | 19.8v 1987 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
3 | 1, 2 | impbii 211 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 |
This theorem depends on definitions: df-bi 209 df-ex 1781 |
This theorem is referenced by: 19.3vOLD 1989 19.36v 1994 19.44v 1999 19.45v 2000 zfcndpow 10038 volfiniune 31489 bnj937 32043 bnj594 32184 bnj907 32239 bnj1128 32262 bnj1145 32265 coss0 35734 prter2 36032 relopabVD 41255 rfcnnnub 41313 |
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