Theorem rspec2 2566

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 2529	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)
3	2	r19.21bi 2565	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2148  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-4 1510

This theorem depends on definitions:  df-bi 117  df-ral 2460

This theorem is referenced by:  rspec3  2567  ordtriexmid  4519  onsucsssucexmid  4525