Theorem bi2raldva 1233

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypothesis

Ref	Expression
bi2raldva.1	⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))

Assertion

Ref	Expression
bi2raldva	⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

Distinct variable group(s):   x,y,φ   y,A

Proof of Theorem bi2raldva

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (φ → ∀xφ)
2		ax-17 925	. 2 ⊢ (φ → ∀yφ)
3		bi2raldva.1	. 2 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))
4	1, 2, 3	bi2ralda 1232	1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  f1oweOLD 2944

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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