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Theorem gencbval 2903
 Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
gencbval.1 A V
gencbval.2 (A = y → (φψ))
gencbval.3 (A = y → (χθ))
gencbval.4 (θx(χ A = y))
Assertion
Ref Expression
gencbval (x(χφ) ↔ y(θψ))
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 gencbval
StepHypRef Expression
1 gencbval.1 . . . 4 A V
2 gencbval.2 . . . . 5 (A = y → (φψ))
32notbid 285 . . . 4 (A = y → (¬ φ ↔ ¬ ψ))
4 gencbval.3 . . . 4 (A = y → (χθ))
5 gencbval.4 . . . 4 (θx(χ A = y))
61, 3, 4, 5gencbvex 2901 . . 3 (x(χ ¬ φ) ↔ y(θ ¬ ψ))
7 exanali 1585 . . 3 (x(χ ¬ φ) ↔ ¬ x(χφ))
8 exanali 1585 . . 3 (y(θ ¬ ψ) ↔ ¬ y(θψ))
96, 7, 83bitr3i 266 . 2 x(χφ) ↔ ¬ y(θψ))
109con4bii 288 1 (x(χφ) ↔ y(θψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 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-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)
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