Theorem rspec2 2450

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

Hypothesis

Ref	Expression
rspec2.1	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Assertion

Ref	Expression
rspec2	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)

Proof of Theorem rspec2

Step	Hyp	Ref	Expression
1		rspec2.1	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
2	1	rspec 2415	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)
3	2	r19.21bi 2449	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1433  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-4 1440

This theorem depends on definitions:  df-bi 115  df-ral 2353

This theorem is referenced by:  rspec3  2451  ordtriexmid  4265  onsucsssucexmid  4270