Theorem 19.9v 2064

Description: Version of 19.9 2227 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 2065. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2092. (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 1992	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 2063	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 199	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∃wex 1851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056

This theorem depends on definitions:  df-bi 197  df-ex 1852

This theorem is referenced by:  19.3v  2065  19.23vOLD  2070  19.36v  2071  19.44v  2079  19.45v  2080  19.41vOLD  2081  zfcndpow  9639  volfiniune  30627  bnj937  31174  bnj594  31314  bnj907  31367  bnj1128  31390  bnj1145  31393  bj-sbfvv  33095  coss0  34564  prter2  34682  relopabVD  39653  rfcnnnub  39711