Theorem rspec3

Description: Specialization rule for restricted quantification, with three quantifiers. (Contributed by NM, 20-Nov-1994)

	Ref	Expression
Hypothesis	rspec3.1	⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑
Assertion	rspec3	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) → 𝜑 )

Proof

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