Theorem 19.9v 1988

Description: Version of 19.9 2203 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.)

Assertion

Ref	Expression
19.9v	⊢ (∃𝑥𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem 19.9v

Step	Hyp	Ref	Expression
1		ax5e 1913	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1987	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∃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 1911  ax-6 1970

This theorem depends on definitions:  df-bi 210  df-ex 1782

This theorem is referenced by:  19.3vOLD  1989  19.36v  1994  19.44v  1999  19.45v  2000  zfcndpow  10027  volfiniune  31599  bnj937  32153  bnj594  32294  bnj907  32349  bnj1128  32372  bnj1145  32375  coss0  35879  prter2  36177  relopabVD  41607  rfcnnnub  41665