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Theorem rspc2ev 2963
 Description: 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
Hypotheses
Ref Expression
rspc2v.1 (x = A → (φχ))
rspc2v.2 (y = B → (χψ))
Assertion
Ref Expression
rspc2ev ((A C B D ψ) → x C y D φ)
Distinct variable groups:   x,y,A   y,B   x,C   x,D,y   χ,x   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x)   χ(y)   B(x)   C(y)

Proof of Theorem rspc2ev
StepHypRef Expression
1 rspc2v.2 . . . . 5 (y = B → (χψ))
21rspcev 2955 . . . 4 ((B D ψ) → y D χ)
32anim2i 552 . . 3 ((A C (B D ψ)) → (A C y D χ))
433impb 1147 . 2 ((A C B D ψ) → (A C y D χ))
5 rspc2v.1 . . . 4 (x = A → (φχ))
65rexbidv 2635 . . 3 (x = A → (y D φy D χ))
76rspcev 2955 . 2 ((A C y D χ) → x C y D φ)
84, 7syl 15 1 ((A C B D ψ) → x C y D 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-1 5  ax-2 6  ax-3 7  ax-mp 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:  rspc3ev  2965  eladdci  4399  rspceov  5556  nclec  6195  ltcpw1pwg  6202  nc0le1  6216  nclenc  6222  ce2le  6233  tlenc1c  6240
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