Theorem rsp2 3279

Description: Restricted specialization, with two quantifiers. (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 3250	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑))
2		rsp 3250	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑))
3	1, 2	syl6 35	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)))
4	3	impd 414	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  ∀wral 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-ral 3077

This theorem is referenced by:  ralcom2  3364  disjxiun  5097  mpocurryd  8249  cmncom  19838  cnmpt21  23731  cnmpt2t  23733  cnmpt22  23734  cnmptcom  23738  frgrwopreglem5ALT  30524  htthlem  31120  qsidomlem2  33640  cplgredgex  35471  disjimeceqim2  39304  eldisjim3  39314  disjlem14  39400  prtlem14  39498  islptre  46195  sprsymrelfolem2  48099