Theorem 19.9v 1991

Description: Version of 19.9 2217 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 1989. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2015. (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 1919	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1990	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 210	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 207  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974

This theorem depends on definitions:  df-bi 208  df-ex 1787

This theorem is referenced by:  19.36v  2000  19.44v  2005  19.45v  2006  zfcndpow  10530  volfiniune  34414  bnj937  34954  bnj594  35094  bnj907  35149  bnj1128  35172  bnj1145  35175  coss0  38936  prter2  39373  relopabVD  45344  rfcnnnub  45484