Theorem 2ralbidva 2340

Description: Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)

Hypothesis

Ref	Expression
2ralbidva.1	⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))

Assertion

Ref	Expression
2ralbidva	⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

Distinct variable groups:   x,y,φ   y,A

Allowed substitution hints:   ψ(x,y)   χ(x,y)   A(x)   B(x,y)

Proof of Theorem 2ralbidva

Step	Hyp	Ref	Expression
1		nfv 1418	. 2 ⊢ Ⅎxφ
2		nfv 1418	. 2 ⊢ Ⅎyφ
3		2ralbidva.1	. 2 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))
4	1, 2, 3	2ralbida 2339	1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∀wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305

This theorem is referenced by:  soinxp  4353  isotr  5399