Theorem rspec3 3284

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

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 3283	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑)
3	2	r19.21bi 3256	. 2 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑)
4	3	3impa 1127	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-ral 3079

This theorem is used by: (None)