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Metamath Proof Explorer

Theorem 2rexbidva

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004)

Ref Expression
Hypothesis 2ralbidva.1 ⊢ φ ∧ x ∈ A ∧ y ∈ B → ψ ↔ χ
Assertion 2rexbidva ⊢ φ → ∃ x ∈ A ∃ y ∈ B ψ ↔ ∃ x ∈ A ∃ y ∈ B χ

Proof

Step Hyp Ref Expression
1 2ralbidva.1 ⊢ φ ∧ x ∈ A ∧ y ∈ B → ψ ↔ χ
2 1 anassrs ⊢ φ ∧ x ∈ A ∧ y ∈ B → ψ ↔ χ
3 2 rexbidva ⊢ φ ∧ x ∈ A → ∃ y ∈ B ψ ↔ ∃ y ∈ B χ
4 3 rexbidva ⊢ φ → ∃ x ∈ A ∃ y ∈ B ψ ↔ ∃ x ∈ A ∃ y ∈ B χ