Theorem 19.9v 2016

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 2014. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2040. (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 1944	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 2015	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  19.36v  2025  19.44v  2030  19.45v  2031  zfcndpow  10706  volfiniune  34911  bnj937  35450  bnj594  35590  bnj907  35645  bnj1128  35668  bnj1145  35671  coss0  39608  prter2  40045  relopabVD  45990  rfcnnnub  46160