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Theorem sbcied 2772
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied.1 (φA 𝑉)
sbcied.2 ((φ x = A) → (ψχ))
Assertion
Ref Expression
sbcied (φ → ([A / x]ψχ))
Distinct variable groups:   x,A   φ,x   χ,x
Allowed substitution hints:   ψ(x)   𝑉(x)

Proof of Theorem sbcied
StepHypRef Expression
1 sbcied.1 . 2 (φA 𝑉)
2 sbcied.2 . 2 ((φ x = A) → (ψχ))
3 nfv 1398 . 2 xφ
4 nfvd 1399 . 2 (φ → Ⅎxχ)
51, 2, 3, 4sbciedf 2771 1 (φ → ([A / x]ψχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226   wcel 1370  [wsbc 2737
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-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-sbc 2738
This theorem is referenced by:  sbcied2  2773  sbc2iedv  2803  sbc3ie  2804  sbcralt  2807  sbcrext  2808  euotd  3961  riota5f  5412
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