Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rspc2vd Structured version   Visualization version   GIF version

Theorem rspc2vd 41437
 Description: Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class 𝐷 for the second set variable 𝑦 may depend on the first set variable 𝑥. (Contributed by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
rspc2vd.a (𝑥 = 𝐴 → (𝜃𝜒))
rspc2vd.b (𝑦 = 𝐵 → (𝜒𝜓))
rspc2vd.c (𝜑𝐴𝐶)
rspc2vd.d ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐸)
rspc2vd.e (𝜑𝐵𝐸)
Assertion
Ref Expression
rspc2vd (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃𝜓))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶   𝑦,𝐷   𝑥,𝐸   𝜑,𝑥   𝜒,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥)   𝐸(𝑦)

Proof of Theorem rspc2vd
StepHypRef Expression
1 rspc2vd.e . . 3 (𝜑𝐵𝐸)
2 rspc2vd.c . . . 4 (𝜑𝐴𝐶)
3 rspc2vd.d . . . 4 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐸)
42, 3csbied 3526 . . 3 (𝜑𝐴 / 𝑥𝐷 = 𝐸)
51, 4eleqtrrd 2691 . 2 (𝜑𝐵𝐴 / 𝑥𝐷)
6 nfcsb1v 3515 . . . . 5 𝑥𝐴 / 𝑥𝐷
7 nfv 1830 . . . . 5 𝑥𝜒
86, 7nfral 2929 . . . 4 𝑥𝑦 𝐴 / 𝑥𝐷𝜒
9 csbeq1a 3508 . . . . 5 (𝑥 = 𝐴𝐷 = 𝐴 / 𝑥𝐷)
10 rspc2vd.a . . . . 5 (𝑥 = 𝐴 → (𝜃𝜒))
119, 10raleqbidv 3129 . . . 4 (𝑥 = 𝐴 → (∀𝑦𝐷 𝜃 ↔ ∀𝑦 𝐴 / 𝑥𝐷𝜒))
128, 11rspc 3276 . . 3 (𝐴𝐶 → (∀𝑥𝐶𝑦𝐷 𝜃 → ∀𝑦 𝐴 / 𝑥𝐷𝜒))
132, 12syl 17 . 2 (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃 → ∀𝑦 𝐴 / 𝑥𝐷𝜒))
14 rspc2vd.b . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
1514rspcv 3278 . 2 (𝐵𝐴 / 𝑥𝐷 → (∀𝑦 𝐴 / 𝑥𝐷𝜒𝜓))
165, 13, 15sylsyld 59 1 (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-sbc 3403  df-csb 3500 This theorem is referenced by:  frcond1  41438
 Copyright terms: Public domain W3C validator