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Theorem rspc3ev 2965
 Description: 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
rspc3v.1 (x = A → (φχ))
rspc3v.2 (y = B → (χθ))
rspc3v.3 (z = C → (θψ))
Assertion
Ref Expression
rspc3ev (((A R B S C T) ψ) → x R y S z T φ)
Distinct variable groups:   ψ,z   χ,x   θ,y   x,y,z,A   y,B,z   z,C   x,R   x,S,y   x,T,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y)   χ(y,z)   θ(x,z)   B(x)   C(x,y)   R(y,z)   S(z)

Proof of Theorem rspc3ev
StepHypRef Expression
1 simpl1 958 . 2 (((A R B S C T) ψ) → A R)
2 simpl2 959 . 2 (((A R B S C T) ψ) → B S)
3 rspc3v.3 . . . 4 (z = C → (θψ))
43rspcev 2955 . . 3 ((C T ψ) → z T θ)
543ad2antl3 1119 . 2 (((A R B S C T) ψ) → z T θ)
6 rspc3v.1 . . . 4 (x = A → (φχ))
76rexbidv 2635 . . 3 (x = A → (z T φz T χ))
8 rspc3v.2 . . . 4 (y = B → (χθ))
98rexbidv 2635 . . 3 (y = B → (z T χz T θ))
107, 9rspc2ev 2963 . 2 ((A R B S z T θ) → x R y S z T φ)
111, 2, 5, 10syl3anc 1182 1 (((A R B S C T) ψ) → x R y S z T φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 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-3an 936  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-rex 2620  df-v 2861 This theorem is referenced by: (None)
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