Theorem rsp2 2547

Description: Restricted specialization. (Contributed by NM, 11-Feb-1997.)

Assertion

Ref	Expression
rsp2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Proof of Theorem rsp2

Step	Hyp	Ref	Expression
1		rsp 2544	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑))
2		rsp 2544	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑))
3	1, 2	syl6 33	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)))
4	3	impd 254	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2167  ∀wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-4 1524

This theorem depends on definitions:  df-bi 117  df-ral 2480

This theorem is referenced by:  ralidm  3551  sowlin  4355  cmncom  13432  cnmpt21  14527  cnmpt2t  14529  cnmpt22  14530  cnmptcom  14534