MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspc2dv Structured version   Visualization version   GIF version

Theorem rspc2dv 3623
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 6-Mar-2025.)
Hypotheses
Ref Expression
rspc2dv.1 (𝑥 = 𝐴 → (𝜓𝜃))
rspc2dv.2 (𝑦 = 𝐵 → (𝜃𝜒))
rspc2dv.3 (𝜑 → ∀𝑥𝐶𝑦𝐷 𝜓)
rspc2dv.4 (𝜑𝐴𝐶)
rspc2dv.5 (𝜑𝐵𝐷)
Assertion
Ref Expression
rspc2dv (𝜑𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷,𝑦   𝜒,𝑦   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥)   𝜃(𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem rspc2dv
StepHypRef Expression
1 rspc2dv.4 . 2 (𝜑𝐴𝐶)
2 rspc2dv.5 . 2 (𝜑𝐵𝐷)
3 rspc2dv.3 . 2 (𝜑 → ∀𝑥𝐶𝑦𝐷 𝜓)
4 rspc2dv.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜃))
5 rspc2dv.2 . . 3 (𝑦 = 𝐵 → (𝜃𝜒))
64, 5rspc2va 3620 . 2 (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜓) → 𝜒)
71, 2, 3, 6syl21anc 836 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062
This theorem is referenced by:  mulscom  27524  addsdilem3  27537  addsdilem4  27538  mulsasslem3  27549
  Copyright terms: Public domain W3C validator