Theorem sbcal 3103

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 3060	. 2 |- ( [. A / y ]. A. x ph -> A e. _V )
2		sbcex 3060	. . 3 |- ( [. A / y ]. ph -> A e. _V )
3	2	sps 1590	. 2 |- ( A. x [. A / y ]. ph -> A e. _V )
4		dfsbcq2 3054	. . 3 |- ( z = A -> ( [ z / y ] A. x ph <-> [. A / y ]. A. x ph ) )
5		dfsbcq2 3054	. . . 4 |- ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6	5	albidv 1877	. . 3 |- ( z = A -> ( A. x [ z / y ] ph <-> A. x [. A / y ]. ph ) )
7		sbal 2060	. . 3 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
8	4, 6, 7	vtoclbg 2884	. 2 |- ( A e. _V -> ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph ) )
9	1, 3, 8	pm5.21nii 716	1 |- ( [. A / y ]. A. x ph <-> A. x [. A / y ]. ph )

Syntax hints:   <-> wb 105   A.wal 1400   = wceq 1402   [wsb 1815   e. wcel 2209   _Vcvv 2821   [.wsbc 3051

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052

This theorem is referenced by:  sbcfung  5396