Theorem sbcal 2997

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 2954	. 2 |- ( [. A / y ]. A. x ph -> A e. _V )
2		sbcex 2954	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	sps 1524	. 2 |- ( A. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 2949	. . 3 |- ( z = A -> ( [ z / y ] A. x ph <-> [. A / y ]. A. x ph ) )
5		dfsbcq2 2949	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	albidv 1811	. . 3 |- ( z = A -> ( A. x [ z / y ] ph <-> A. x [. A / y ]. ph ) )
7		sbal 1987	. . 3 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2782	. 2 |- ( A e. _V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 694	1 |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )

Syntax hints:   <-> wb 104   A.wal 1340   = wceq 1342   [wsb 1749   e. wcel 2135   _Vcvv 2721   [.wsbc 2946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-sbc 2947

This theorem is referenced by:  sbcfung  5206