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Theorem rspc2 3569
Description: Restricted specialization with two quantifiers, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
Hypotheses
Ref Expression
rspc2.1 𝑥𝜒
rspc2.2 𝑦𝜓
rspc2.3 (𝑥 = 𝐴 → (𝜑𝜒))
rspc2.4 (𝑦 = 𝐵 → (𝜒𝜓))
Assertion
Ref Expression
rspc2 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem rspc2
StepHypRef Expression
1 nfcv 2909 . . . 4 𝑥𝐷
2 rspc2.1 . . . 4 𝑥𝜒
31, 2nfralw 3152 . . 3 𝑥𝑦𝐷 𝜒
4 rspc2.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
54ralbidv 3123 . . 3 (𝑥 = 𝐴 → (∀𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 𝜒))
63, 5rspc 3548 . 2 (𝐴𝐶 → (∀𝑥𝐶𝑦𝐷 𝜑 → ∀𝑦𝐷 𝜒))
7 rspc2.2 . . 3 𝑦𝜓
8 rspc2.4 . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
97, 8rspc 3548 . 2 (𝐵𝐷 → (∀𝑦𝐷 𝜒𝜓))
106, 9sylan9 508 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wnf 1790  wcel 2110  wral 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-v 3433
This theorem is referenced by:  reu2eqd  3675  reuop  6195  fvmpocurryd  8079  dvmptfsum  25150  poimirlem26  35812  fphpd  40647  reupr  44953
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