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Theorem gencl 2887
 Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
gencl.1 (θx(χ A = B))
gencl.2 (A = B → (φψ))
gencl.3 (χφ)
Assertion
Ref Expression
gencl (θψ)
Distinct variable group:   ψ,x
Allowed substitution hints:   φ(x)   χ(x)   θ(x)   A(x)   B(x)

Proof of Theorem gencl
StepHypRef Expression
1 gencl.1 . 2 (θx(χ A = B))
2 gencl.3 . . . . 5 (χφ)
3 gencl.2 . . . . 5 (A = B → (φψ))
42, 3syl5ib 210 . . . 4 (A = B → (χψ))
54impcom 419 . . 3 ((χ A = B) → ψ)
65exlimiv 1634 . 2 (x(χ A = B) → ψ)
71, 6sylbi 187 1 (θψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  2gencl  2888  3gencl  2889
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