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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-mp 5  ax-1 6  ax-2 7  ax-3 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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