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Theorem rspc2 2852
Description: 2-variable restricted specialization, 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 2319 . . . 4 𝑥𝐷
2 rspc2.1 . . . 4 𝑥𝜒
31, 2nfralxy 2515 . . 3 𝑥𝑦𝐷 𝜒
4 rspc2.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
54ralbidv 2477 . . 3 (𝑥 = 𝐴 → (∀𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 𝜒))
63, 5rspc 2835 . 2 (𝐴𝐶 → (∀𝑥𝐶𝑦𝐷 𝜑 → ∀𝑦𝐷 𝜒))
7 rspc2.2 . . 3 𝑦𝜓
8 rspc2.4 . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
97, 8rspc 2835 . 2 (𝐵𝐷 → (∀𝑦𝐷 𝜒𝜓))
106, 9sylan9 409 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wnf 1460  wcel 2148  wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739
This theorem is referenced by:  rspc2v  2854  disjiun  3997
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