Theorem sbcal 3029

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 2986	. 2 |- ( [. A / y ]. A. x ph -> A e. _V )
2		sbcex 2986	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	sps 1548	. 2 |- ( A. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 2980	. . 3 |- ( z = A -> ( [ z / y ] A. x ph <-> [. A / y ]. A. x ph ) )
5		dfsbcq2 2980	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	albidv 1835	. . 3 |- ( z = A -> ( A. x [ z / y ] ph <-> A. x [. A / y ]. ph ) )
7		sbal 2012	. . 3 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2813	. 2 |- ( A e. _V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 705	1 |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   [wsb 1773   e. wcel 2160   _Vcvv 2752   [.wsbc 2977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978

This theorem is referenced by:  sbcfung  5259