Theorem sbcal 2888

Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)

Assertion

Ref	Expression
sbcal	|- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )

Distinct variable groups:   x, A   x, y

Allowed substitution hints:   ph( x, y)   A( y)

Proof of Theorem sbcal

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 2846	. 2 |- ( [. A / y ]. A. x ph -> A e. _V )
2		sbcex 2846	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	sps 1475	. 2 |- ( A. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 2841	. . 3 |- ( z = A -> ( [ z / y ] A. x ph <-> [. A / y ]. A. x ph ) )
5		dfsbcq2 2841	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	albidv 1752	. . 3 |- ( z = A -> ( A. x [ z / y ] ph <-> A. x [. A / y ]. ph ) )
7		sbal 1924	. . 3 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2680	. 2 |- ( A e. _V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 655	1 |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )

Syntax hints:   <-> wb 103   A.wal 1287   = wceq 1289   e. wcel 1438   [wsb 1692   _Vcvv 2619   [.wsbc 2838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839

This theorem is referenced by:  sbcfung  5025