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Theorem rspc2 2960
 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 ((A C B D) → (x C y D 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 2489 . . . 4 xD
2 rspc2.1 . . . 4 xχ
31, 2nfral 2667 . . 3 xy D χ
4 rspc2.3 . . . 4 (x = A → (φχ))
54ralbidv 2634 . . 3 (x = A → (y D φy D χ))
63, 5rspc 2949 . 2 (A C → (x C y D φy D χ))
7 rspc2.2 . . 3 yψ
8 rspc2.4 . . 3 (y = B → (χψ))
97, 8rspc 2949 . 2 (B D → (y D χψ))
106, 9sylan9 638 1 ((A C B D) → (x C y D φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861 This theorem is referenced by:  rspc2v  2961
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