Theorem 19.9v 1984

Description: Version of 19.9 2200 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 1982. (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 1909	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1983	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 211	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by:  19.3vOLD  1985  19.36v  1990  19.44v  1995  19.45v  1996  zfcndpow  10032  volfiniune  31484  bnj937  32038  bnj594  32179  bnj907  32234  bnj1128  32257  bnj1145  32260  coss0  35713  prter2  36011  relopabVD  41228  rfcnnnub  41286