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Theorem sbiedv 2037
 Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2038). (Contributed by NM, 7-Jan-2017.)
Hypothesis
Ref Expression
sbiedv.1 ((φ x = y) → (ψχ))
Assertion
Ref Expression
sbiedv (φ → ([y / x]ψχ))
Distinct variable groups:   φ,x   χ,x
Allowed substitution hints:   φ(y)   ψ(x,y)   χ(y)

Proof of Theorem sbiedv
StepHypRef Expression
1 nfv 1619 . 2 xφ
2 nfvd 1620 . 2 (φ → Ⅎxχ)
3 sbiedv.1 . . 3 ((φ x = y) → (ψχ))
43ex 423 . 2 (φ → (x = y → (ψχ)))
51, 2, 4sbied 2036 1 (φ → ([y / x]ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  [wsb 1648 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-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by: (None)
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