Theorem rsp2 2677

Description: Restricted specialization. (Contributed by NM, 11-Feb-1997.)

Assertion

Ref	Expression
rsp2	⊢ (∀x ∈ A ∀y ∈ B φ → ((x ∈ A ∧ y ∈ B) → φ))

Proof of Theorem rsp2

Step	Hyp	Ref	Expression
1		rsp 2675	. . 3 ⊢ (∀x ∈ A ∀y ∈ B φ → (x ∈ A → ∀y ∈ B φ))
2		rsp 2675	. . 3 ⊢ (∀y ∈ B φ → (y ∈ B → φ))
3	1, 2	syl6 29	. 2 ⊢ (∀x ∈ A ∀y ∈ B φ → (x ∈ A → (y ∈ B → φ)))
4	3	imp3a 420	1 ⊢ (∀x ∈ A ∀y ∈ B φ → ((x ∈ A ∧ y ∈ B) → φ))

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀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-ex 1542  df-ral 2620

This theorem is referenced by:  ralcom2  2776  f1fveq  5474