Theorem 2ralbida 2339

Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)

Hypotheses

Ref	Expression
2ralbida.1	⊢ Ⅎxφ
2ralbida.2	⊢ Ⅎyφ
2ralbida.3	⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))

Assertion

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

Distinct variable groups:   x,y   y,A

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

Proof of Theorem 2ralbida

Step	Hyp	Ref	Expression
1		2ralbida.1	. 2 ⊢ Ⅎxφ
2		2ralbida.2	. . . 4 ⊢ Ⅎyφ
3		nfv 1418	. . . 4 ⊢ Ⅎy x ∈ A
4	2, 3	nfan 1454	. . 3 ⊢ Ⅎy(φ ∧ x ∈ A)
5		2ralbida.3	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))
6	5	anassrs 380	. . 3 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ B) → (ψ ↔ χ))
7	4, 6	ralbida 2314	. 2 ⊢ ((φ ∧ x ∈ A) → (∀y ∈ B ψ ↔ ∀y ∈ B χ))
8	1, 7	ralbida 2314	1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  Ⅎwnf 1346   ∈ 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:  2ralbidva  2340