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Theorem sbcal 2787
 Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcal ([A / y]xφx[A / y]φ)
Distinct variable groups:   x,A   x,y
Allowed substitution hints:   φ(x,y)   A(y)

Proof of Theorem sbcal
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbcex 2749 . 2 ([A / y]xφA V)
2 sbcex 2749 . . 3 ([A / y]φA V)
32sps 1412 . 2 (x[A / y]φA V)
4 dfsbcq2 2744 . . 3 (z = A → ([z / y]xφ[A / y]xφ))
5 dfsbcq2 2744 . . . 4 (z = A → ([z / y]φ[A / y]φ))
65albidv 1687 . . 3 (z = A → (x[z / y]φx[A / y]φ))
7 sbal 1858 . . 3 ([z / y]xφx[z / y]φ)
84, 6, 7vtoclbg 2591 . 2 (A V → ([A / y]xφx[A / y]φ))
91, 3, 8pm5.21nii 607 1 ([A / y]xφx[A / y]φ)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  [wsb 1627  Vcvv 2535  [wsbc 2741 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742 This theorem is referenced by:  sbcfung  4851
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