Theorem rspec3 1268

Description: Specialization rule for restricted quantification.

Hypothesis

Ref	Expression
rspec3.1	⊢ ∀x ∈ A ∀y ∈ B ∀z ∈ C φ

Assertion

Ref	Expression
rspec3	⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ)

Proof of Theorem rspec3

Step	Hyp	Ref	Expression
1		rspec3.1	. . . 4 ⊢ ∀x ∈ A ∀y ∈ B ∀z ∈ C φ
2	1	rspec2 1267	. . 3 ⊢ ((x ∈ A ∧ y ∈ B) → ∀z ∈ C φ)
3	2	r19.21bi 1266	. 2 ⊢ (((x ∈ A ∧ y ∈ B) ∧ z ∈ C) → φ)
4	3	3impa 609	1 ⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ)

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581   ∈ wcel 1092  ∀wral 1201

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-3an 583  df-ral 1205

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