This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem rspec2

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

Ref Expression
Hypothesis rspec2.1 ⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑
Assertion rspec2 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 )

Proof

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