Theorem ra42 1245

Description: Restricted specialization.

Assertion

Ref	Expression
ra42	⊢ (∀x ∈ A ∀y ∈ B φ → ((x ∈ A ∧ y ∈ B) → φ))

Proof of Theorem ra42

Step	Hyp	Ref	Expression
1		ra4 1243	. . 3 ⊢ (∀x ∈ A ∀y ∈ B φ → (x ∈ A → ∀y ∈ B φ))
2		ra4 1243	. . 3 ⊢ (∀y ∈ B φ → (y ∈ B → φ))
3	1, 2	syl6 23	. 2 ⊢ (∀x ∈ A ∀y ∈ B φ → (x ∈ A → (y ∈ B → φ)))
4	3	imp3a 279	1 ⊢ (∀x ∈ A ∀y ∈ B φ → ((x ∈ A ∧ y ∈ B) → φ))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  solin 2145  ralxp 2456  f1fveq 2918  isotrALT 2936

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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