Theorem 19.9v 1896

Description: Version of 19.9 2072 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1897. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1935. (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 1841	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		19.8v 1895	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbii 199	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  19.3v  1897  19.23v  1902  19.36v  1904  19.44v  1912  19.45v  1913  19.41v  1914  elsnxpOLD  5678  zfcndpow  9438  volfiniune  30293  bnj937  30842  bnj594  30982  bnj907  31035  bnj1128  31058  bnj1145  31061  bj-sbfvv  32765  prter2  34166  relopabVD  39137  rfcnnnub  39195