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Theorem sbcal 2804
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 2766 . 2 ([A / y]xφA V)
2 sbcex 2766 . . 3 ([A / y]φA V)
32sps 1427 . 2 (x[A / y]φA V)
4 dfsbcq2 2761 . . 3 (z = A → ([z / y]xφ[A / y]xφ))
5 dfsbcq2 2761 . . . 4 (z = A → ([z / y]φ[A / y]φ))
65albidv 1702 . . 3 (z = A → (x[z / y]φx[A / y]φ))
7 sbal 1873 . . 3 ([z / y]xφx[z / y]φ)
84, 6, 7vtoclbg 2608 . 2 (A V → ([A / y]xφx[A / y]φ))
91, 3, 8pm5.21nii 619 1 ([A / y]xφx[A / y]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = wceq 1242   wcel 1390  [wsb 1642  Vcvv 2551  [wsbc 2758
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  sbcfung  4868
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