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Theorem rspc2 3616
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 2891 . . . 4 𝑥𝐷
2 rspc2.1 . . . 4 𝑥𝜒
31, 2nfralw 3298 . . 3 𝑥𝑦𝐷 𝜒
4 rspc2.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
54ralbidv 3167 . . 3 (𝑥 = 𝐴 → (∀𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 𝜒))
63, 5rspc 3595 . 2 (𝐴𝐶 → (∀𝑥𝐶𝑦𝐷 𝜑 → ∀𝑦𝐷 𝜒))
7 rspc2.2 . . 3 𝑦𝜓
8 rspc2.4 . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
97, 8rspc 3595 . 2 (𝐵𝐷 → (∀𝑦𝐷 𝜒𝜓))
106, 9sylan9 506 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wnf 1777  wcel 2098  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-v 3463
This theorem is referenced by:  reu2eqd  3729  reuop  6303  fvmpocurryd  8285  dvmptfsum  25990  poimirlem26  37307  fphpd  42422  reupr  47043
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