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

Theorem rsp2 3275
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 3245 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 rsp 3245 . . 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 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-ral 3060
This theorem is referenced by:  ralcom2  3375  disjxiun  5145  mpocurryd  8293  cmncom  19831  cnmpt21  23695  cnmpt2t  23697  cnmpt22  23698  cnmptcom  23702  frgrwopreglem5ALT  30351  htthlem  30946  qsidomlem2  33461  cplgredgex  35105  disjlem14  38780  prtlem14  38856  islptre  45575  sprsymrelfolem2  47418
  Copyright terms: Public domain W3C validator