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Theorem sbied 1649
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1652). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypotheses
Ref Expression
sbied.1 xφ
sbied.2 (φ → Ⅎxχ)
sbied.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
sbied (φ → ([y / x]ψχ))

Proof of Theorem sbied
StepHypRef Expression
1 sbied.1 . . 3 xφ
21nfri 1389 . 2 (φxφ)
3 sbied.2 . . 3 (φ → Ⅎxχ)
43nfrd 1390 . 2 (φ → (χxχ))
5 sbied.3 . 2 (φ → (x = y → (ψχ)))
62, 4, 5sbiedh 1648 1 (φ → ([y / x]ψχ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1325  [wsb 1623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-i9 1400  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624
This theorem is referenced by:  sbiedv  1650  dvelimdf  1870  cbvrald  7226
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