New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  spc2egv GIF version

Theorem spc2egv 2941
 Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
Hypothesis
Ref Expression
spc2egv.1 ((x = A y = B) → (φψ))
Assertion
Ref Expression
spc2egv ((A V B W) → (ψxyφ))
Distinct variable groups:   x,y,A   x,B,y   ψ,x,y
Allowed substitution hints:   φ(x,y)   V(x,y)   W(x,y)

Proof of Theorem spc2egv
StepHypRef Expression
1 elisset 2869 . . . 4 (A Vx x = A)
2 elisset 2869 . . . 4 (B Wy y = B)
31, 2anim12i 549 . . 3 ((A V B W) → (x x = A y y = B))
4 eeanv 1913 . . 3 (xy(x = A y = B) ↔ (x x = A y y = B))
53, 4sylibr 203 . 2 ((A V B W) → xy(x = A y = B))
6 spc2egv.1 . . . 4 ((x = A y = B) → (φψ))
76biimprcd 216 . . 3 (ψ → ((x = A y = B) → φ))
872eximdv 1624 . 2 (ψ → (xy(x = A y = B) → xyφ))
95, 8syl5com 26 1 ((A V B W) → (ψxyφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710 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-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  spc2gv  2942  spc2ev  2947
 Copyright terms: Public domain W3C validator