Theorem 19.9v 2014

Description: Version of 19.9 2241 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 2012. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2038. (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 1942	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 2013	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  19.36v  2023  19.44v  2028  19.45v  2029  zfcndpow  10596  volfiniune  34620  bnj937  35160  bnj594  35300  bnj907  35355  bnj1128  35378  bnj1145  35381  coss0  39218  prter2  39655  relopabVD  45609  rfcnnnub  45756