Theorem 19.9v 1985

Description: Version of 19.9 2208 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 1983. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2009. (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 1984	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 209	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∃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 1968

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by:  19.36v  1994  19.44v  1999  19.45v  2000  zfcndpow  10502  volfiniune  34235  bnj937  34775  bnj594  34916  bnj907  34971  bnj1128  34994  bnj1145  34997  coss0  38516  prter2  38920  relopabVD  44933  rfcnnnub  45073