Theorem 19.9v 2019

Description: Version of 19.9 2245 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 2017. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2043. (Revised by Wolf Lammen, 4-Dec-2017.)

Assertion

Ref	Expression
19.9v	⊢ (∃𝑥𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem 19.9v

Step	Hyp	Ref	Expression
1		ax5e 1947	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 2018	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  19.36v  2028  19.44v  2033  19.45v  2034  zfcndpow  10740  volfiniune  34974  bnj937  35513  bnj594  35653  bnj907  35708  bnj1128  35731  bnj1145  35734  coss0  39767  prter2  40204  relopabVD  46143  rfcnnnub  46327