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Theorem rspc2 3508
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 2941 . . . 4 𝑥𝐷
2 rspc2.1 . . . 4 𝑥𝜒
31, 2nfral 3126 . . 3 𝑥𝑦𝐷 𝜒
4 rspc2.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
54ralbidv 3167 . . 3 (𝑥 = 𝐴 → (∀𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 𝜒))
63, 5rspc 3491 . 2 (𝐴𝐶 → (∀𝑥𝐶𝑦𝐷 𝜑 → ∀𝑦𝐷 𝜒))
7 rspc2.2 . . 3 𝑦𝜓
8 rspc2.4 . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
97, 8rspc 3491 . 2 (𝐵𝐷 → (∀𝑦𝐷 𝜒𝜓))
106, 9sylan9 504 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wnf 1879  wcel 2157  wral 3089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387
This theorem is referenced by:  rspc2v  3510  reu2eqd  3601  fvmpt2curryd  7635  dvmptfsum  24079  poimirlem26  33924  fphpd  38166
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