Theorem r19.26-2 2748

Description: Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
r19.26-2	⊢ (∀x ∈ A ∀y ∈ B (φ ∧ ψ) ↔ (∀x ∈ A ∀y ∈ B φ ∧ ∀x ∈ A ∀y ∈ B ψ))

Proof of Theorem r19.26-2

Step	Hyp	Ref	Expression
1		r19.26 2747	. . 3 ⊢ (∀y ∈ B (φ ∧ ψ) ↔ (∀y ∈ B φ ∧ ∀y ∈ B ψ))
2	1	ralbii 2639	. 2 ⊢ (∀x ∈ A ∀y ∈ B (φ ∧ ψ) ↔ ∀x ∈ A (∀y ∈ B φ ∧ ∀y ∈ B ψ))
3		r19.26 2747	. 2 ⊢ (∀x ∈ A (∀y ∈ B φ ∧ ∀y ∈ B ψ) ↔ (∀x ∈ A ∀y ∈ B φ ∧ ∀x ∈ A ∀y ∈ B ψ))
4	2, 3	bitri 240	1 ⊢ (∀x ∈ A ∀y ∈ B (φ ∧ ψ) ↔ (∀x ∈ A ∀y ∈ B φ ∧ ∀x ∈ A ∀y ∈ B ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wral 2615

This theorem was proved from 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-11 1746

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

This theorem is referenced by:  fununi  5161