Theorem 19.9v 2011

Description: Version of 19.9 2247 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 2009. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2035. (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 1939	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 2010	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 212	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994

This theorem depends on definitions:  df-bi 210  df-ex 1807

This theorem is referenced by:  19.36v  2020  19.44v  2025  19.45v  2026  zfcndpow  10597  volfiniune  34561  bnj937  35101  bnj594  35241  bnj907  35296  bnj1128  35319  bnj1145  35322  coss0  39103  prter2  39540  relopabVD  45496  rfcnnnub  45643