Theorem 19.9h 1636

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)

Hypothesis

Ref	Expression
19.9h.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
19.9h	⊢ (∃𝑥𝜑 ↔ 𝜑)

Proof of Theorem 19.9h

Step	Hyp	Ref	Expression
1		19.9ht 1634	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
2		19.9h.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1444	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
4		19.8a 1583	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
5	3, 4	impbii 125	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.9  1637  excomim  1656  exdistrfor  1793  sbcof2  1803  ax11ev  1821  19.9v  1864  exists1  2115