Metamath Proof Explorer


Theorem r19.9rzv

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998)

Ref Expression
Assertion r19.9rzv ( 𝐴 ≠ ∅ → ( 𝜑 ↔ ∃ 𝑥𝐴 𝜑 ) )

Proof

Step Hyp Ref Expression
1 dfrex2 ( ∃ 𝑥𝐴 𝜑 ↔ ¬ ∀ 𝑥𝐴 ¬ 𝜑 )
2 r19.3rzv ( 𝐴 ≠ ∅ → ( ¬ 𝜑 ↔ ∀ 𝑥𝐴 ¬ 𝜑 ) )
3 2 con1bid ( 𝐴 ≠ ∅ → ( ¬ ∀ 𝑥𝐴 ¬ 𝜑𝜑 ) )
4 1 3 bitr2id ( 𝐴 ≠ ∅ → ( 𝜑 ↔ ∃ 𝑥𝐴 𝜑 ) )