Theorem 2ralbida 2654

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 1619	. . . 4 ⊢ Ⅎy x ∈ A
4	2, 3	nfan 1824	. . 3 ⊢ Ⅎy(φ ∧ x ∈ A)
5		2ralbida.3	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))
6	5	anassrs 629	. . 3 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ B) → (ψ ↔ χ))
7	4, 6	ralbida 2629	. 2 ⊢ ((φ ∧ x ∈ A) → (∀y ∈ B ψ ↔ ∀y ∈ B χ))
8	1, 7	ralbida 2629	1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is used by:  2ralbidva  2655