Theorem 19.9v 1883

Description: Version of 19.9 2060 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 1884. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1922. (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 1829	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1882	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 198	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 195  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  19.3v  1884  19.23v  1889  19.36v  1891  19.44v  1899  19.45v  1900  19.41v  1901  elsnxpOLD  5595  zfcndpow  9317  volfiniune  29620  bnj937  30096  bnj594  30236  bnj907  30289  bnj1128  30312  bnj1145  30315  bj-sbfvv  31953  prter2  33184  relopabVD  38159  rfcnnnub  38218