Theorem 19.9v 1894

Description: Version of 19.9 2070 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 1895. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1933. (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 1839	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1893	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 199	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∃wex 1702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886

This theorem depends on definitions:  df-bi 197  df-ex 1703

This theorem is referenced by:  19.3v  1895  19.23v  1900  19.36v  1902  19.44v  1910  19.45v  1911  19.41v  1912  elsnxpOLD  5666  zfcndpow  9423  volfiniune  30267  bnj937  30816  bnj594  30956  bnj907  31009  bnj1128  31032  bnj1145  31035  bj-sbfvv  32740  prter2  33985  relopabVD  38957  rfcnnnub  39015