Theorem 19.9v 1953

Description: Version of 19.9 2110 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1954. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1981. (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 1881	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1952	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 199	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∃wex 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945

This theorem depends on definitions:  df-bi 197  df-ex 1745

This theorem is referenced by:  19.3v  1954  19.23vOLD  1959  19.36v  1960  19.44v  1968  19.45v  1969  19.41vOLD  1970  elsnxpOLD  5716  zfcndpow  9476  volfiniune  30421  bnj937  30968  bnj594  31108  bnj907  31161  bnj1128  31184  bnj1145  31187  bj-sbfvv  32890  coss0  34369  prter2  34485  relopabVD  39451  rfcnnnub  39509