Theorem r19.21aivv 1263

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.)

Hypothesis

Ref	Expression
r19.21aivv.1	⊢ (φ → ((x ∈ A ∧ y ∈ B) → ψ))

Assertion

Ref	Expression
r19.21aivv	⊢ (φ → ∀x ∈ A ∀y ∈ B ψ)

Distinct variable group(s):   x,y,φ   y,A

Proof of Theorem r19.21aivv

Step	Hyp	Ref	Expression
1		r19.21aivv.1	. . . 4 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → ψ))
2	1	exp3a 292	. . 3 ⊢ (φ → (x ∈ A → (y ∈ B → ψ)))
3	2	r19.21adv 1262	. 2 ⊢ (φ → (x ∈ A → ∀y ∈ B ψ))
4	3	r19.21aiv 1259	1 ⊢ (φ → ∀x ∈ A ∀y ∈ B ψ)

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

This theorem is referenced by:  dom2d 3307  uzwo2 4606

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

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

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