Theorem 19.9 2072

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1896 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)

Hypothesis

Ref	Expression
19.9.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.9	⊢ (∃𝑥𝜑 ↔ 𝜑)

Proof of Theorem 19.9

Step	Hyp	Ref	Expression
1		19.9.1	. 2 ⊢ Ⅎ𝑥𝜑
2		19.9t 2071	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∃wex 1704  Ⅎwnf 1708

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  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  exlimd  2087  19.19  2097  19.36  2098  19.41  2103  19.44  2106  19.45  2107  19.9h  2120  exists1  2561  dfid3  5025  fsplit  7282  bnj1189  31077  bj-exexbiex  32691  bj-exalbial  32693  ax6e2ndeq  38775  e2ebind  38779  ax6e2ndeqVD  39145  e2ebindVD  39148  e2ebindALT  39165  ax6e2ndeqALT  39167