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Theorem sbiedv 1669
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1671). (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 1418 . 2 xφ
2 nfvd 1419 . 2 (φ → Ⅎxχ)
3 sbiedv.1 . . 3 ((φ x = y) → (ψχ))
43ex 108 . 2 (φ → (x = y → (ψχ)))
51, 2, 4sbied 1668 1 (φ → ([y / x]ψχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  [wsb 1642
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  acexmid  5454
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