MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rsp2 Structured version   Visualization version   GIF version

Theorem rsp2 3273
Description: Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)
Assertion
Ref Expression
rsp2 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))

Proof of Theorem rsp2
StepHypRef Expression
1 rsp 3243 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 rsp 3243 . . 3 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
31, 2syl6 35 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → (𝑦𝐵𝜑)))
43impd 410 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-ral 3061
This theorem is referenced by:  ralcom2  3372  disjxiun  5145  mpocurryd  8260  cmncom  19714  cnmpt21  23495  cnmpt2t  23497  cnmpt22  23498  cnmptcom  23502  frgrwopreglem5ALT  30009  htthlem  30604  qsidomlem2  33013  cplgredgex  34576  disjlem14  38134  prtlem14  38210  islptre  44796  sprsymrelfolem2  46622
  Copyright terms: Public domain W3C validator