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Theorem rspc2 3351
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 2793 . . . 4 𝑥𝐷
2 rspc2.1 . . . 4 𝑥𝜒
31, 2nfral 2974 . . 3 𝑥𝑦𝐷 𝜒
4 rspc2.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
54ralbidv 3015 . . 3 (𝑥 = 𝐴 → (∀𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 𝜒))
63, 5rspc 3334 . 2 (𝐴𝐶 → (∀𝑥𝐶𝑦𝐷 𝜑 → ∀𝑦𝐷 𝜒))
7 rspc2.2 . . 3 𝑦𝜓
8 rspc2.4 . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
97, 8rspc 3334 . 2 (𝐵𝐷 → (∀𝑦𝐷 𝜒𝜓))
106, 9sylan9 690 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233
This theorem is referenced by:  rspc2v  3353  reu2eqd  3436  fvmpt2curryd  7442  dvmptfsum  23783  poimirlem26  33565  fphpd  37697
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