Theorem 19.9v 1987

Description: Version of 19.9 2198 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 1985. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2011. (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 1915	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1986	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 208	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  19.36v  1991  19.44v  1996  19.45v  1997  zfcndpow  10372  volfiniune  32198  bnj937  32751  bnj594  32892  bnj907  32947  bnj1128  32970  bnj1145  32973  coss0  36597  prter2  36895  relopabVD  42521  rfcnnnub  42579