Theorem 19.9v 1986

Description: Version of 19.9 2213 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 1984. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2010. (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 1914	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1985	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 209	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by:  19.36v  1995  19.44v  2000  19.45v  2001  zfcndpow  10539  volfiniune  34408  bnj937  34948  bnj594  35088  bnj907  35143  bnj1128  35166  bnj1145  35169  coss0  38820  prter2  39257  relopabVD  45256  rfcnnnub  45396