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Theorem 2gencl 2888
 Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
2gencl.1 (C Sx R A = C)
2gencl.2 (D Sy R B = D)
2gencl.3 (A = C → (φψ))
2gencl.4 (B = D → (ψχ))
2gencl.5 ((x R y R) → φ)
Assertion
Ref Expression
2gencl ((C S D S) → χ)
Distinct variable groups:   x,y   x,R   ψ,x   y,C   y,S   χ,y
Allowed substitution hints:   φ(x,y)   ψ(y)   χ(x)   A(x,y)   B(x,y)   C(x)   D(x,y)   R(y)   S(x)

Proof of Theorem 2gencl
StepHypRef Expression
1 2gencl.2 . . . 4 (D Sy R B = D)
2 df-rex 2620 . . . 4 (y R B = Dy(y R B = D))
31, 2bitri 240 . . 3 (D Sy(y R B = D))
4 2gencl.4 . . . 4 (B = D → (ψχ))
54imbi2d 307 . . 3 (B = D → ((C Sψ) ↔ (C Sχ)))
6 2gencl.1 . . . . . 6 (C Sx R A = C)
7 df-rex 2620 . . . . . 6 (x R A = Cx(x R A = C))
86, 7bitri 240 . . . . 5 (C Sx(x R A = C))
9 2gencl.3 . . . . . 6 (A = C → (φψ))
109imbi2d 307 . . . . 5 (A = C → ((y Rφ) ↔ (y Rψ)))
11 2gencl.5 . . . . . 6 ((x R y R) → φ)
1211ex 423 . . . . 5 (x R → (y Rφ))
138, 10, 12gencl 2887 . . . 4 (C S → (y Rψ))
1413com12 27 . . 3 (y R → (C Sψ))
153, 5, 14gencl 2887 . 2 (D S → (C Sχ))
1615impcom 419 1 ((C S D S) → χ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 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  df-rex 2620 This theorem is referenced by:  3gencl  2889
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