Theorem 19.9v 2080

Description: Version of 19.9 2239 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 2081. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2107. (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 2008	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 2079	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 201	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 198  ∃wex 1875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072

This theorem depends on definitions:  df-bi 199  df-ex 1876

This theorem is referenced by:  19.3v  2081  19.23vOLD  2086  19.36v  2087  19.44v  2094  19.45v  2095  19.41vOLD  2096  zfcndpow  9726  volfiniune  30809  bnj937  31359  bnj594  31499  bnj907  31552  bnj1128  31575  bnj1145  31578  bj-sbfvv  33263  coss0  34723  prter2  34902  relopabVD  39897  rfcnnnub  39955