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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 ↔ ∃ 𝑥 ∈ 𝐴 𝜑 ) )

Proof

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