Theorem 19.9v 2017

Description: Version of 19.9 2244 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 2015. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2041. (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 1945	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 2016	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  19.36v  2026  19.44v  2031  19.45v  2032  zfcndpow  10612  volfiniune  34661  bnj937  35201  bnj594  35341  bnj907  35396  bnj1128  35419  bnj1145  35422  coss0  39251  prter2  39688  relopabVD  45642  rfcnnnub  45789