New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  gencbvex2 GIF version

Theorem gencbvex2 2902
 Description: Restatement of gencbvex 2901 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)
Hypotheses
Ref Expression
gencbvex2.1 A V
gencbvex2.2 (A = y → (φψ))
gencbvex2.3 (A = y → (χθ))
gencbvex2.4 (θx(χ A = y))
Assertion
Ref Expression
gencbvex2 (x(χ φ) ↔ y(θ ψ))
Distinct variable groups:   ψ,x   φ,y   θ,x   χ,y   y,A
Allowed substitution hints:   φ(x)   ψ(y)   χ(x)   θ(y)   A(x)

Proof of Theorem gencbvex2
StepHypRef Expression
1 gencbvex2.1 . 2 A V
2 gencbvex2.2 . 2 (A = y → (φψ))
3 gencbvex2.3 . 2 (A = y → (χθ))
4 gencbvex2.4 . . 3 (θx(χ A = y))
53biimpac 472 . . . 4 ((χ A = y) → θ)
65exlimiv 1634 . . 3 (x(χ A = y) → θ)
74, 6impbii 180 . 2 (θx(χ A = y))
81, 2, 3, 7gencbvex 2901 1 (x(χ φ) ↔ y(θ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 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: (None)
 Copyright terms: Public domain W3C validator