Theorem sbcal 2928

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 2886	. 2 |- ( [. A / y ]. A. x ph -> A e. _V )
2		sbcex 2886	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	sps 1500	. 2 |- ( A. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 2881	. . 3 |- ( z = A -> ( [ z / y ] A. x ph <-> [. A / y ]. A. x ph ) )
5		dfsbcq2 2881	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	albidv 1778	. . 3 |- ( z = A -> ( A. x [ z / y ] ph <-> A. x [. A / y ]. ph ) )
7		sbal 1951	. . 3 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2718	. 2 |- ( A e. _V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 676	1 |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )

Syntax hints:   <-> wb 104   A.wal 1312   = wceq 1314   e. wcel 1463   [wsb 1718   _Vcvv 2657   [.wsbc 2878

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879

This theorem is referenced by:  sbcfung  5105