This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem rexralbidv

Description: Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006)

Ref Expression
Hypothesis 2ralbidv.1 ⊢ φ → ψ ↔ χ
Assertion rexralbidv ⊢ φ → ∃ x ∈ A ∀ y ∈ B ψ ↔ ∃ x ∈ A ∀ y ∈ B χ

Proof

Step Hyp Ref Expression
1 2ralbidv.1 ⊢ φ → ψ ↔ χ
2 1 ralbidv ⊢ φ → ∀ y ∈ B ψ ↔ ∀ y ∈ B χ
3 2 rexbidv ⊢ φ → ∃ x ∈ A ∀ y ∈ B ψ ↔ ∃ x ∈ A ∀ y ∈ B χ