Theorem sbcan 3074

Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)

Assertion

Ref	Expression
sbcan	|- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )

Proof of Theorem sbcan

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3040	. 2 |- ( [. A / x ]. ( ph /\ ps ) -> A e. _V )
2		sbcex 3040	. . 3 |- ( [. A / x ]. ps -> A e. _V )
3	2	adantl 277	. 2 |- ( ( [. A / x ]. ph /\ [. A / x ]. ps ) -> A e. _V )
4		dfsbcq2 3034	. . 3 |- ( y = A -> ( [ y / x ] ( ph /\ ps ) <-> [. A / x ]. ( ph /\ ps ) ) )
5		dfsbcq2 3034	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		dfsbcq2 3034	. . . 4 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
7	5, 6	anbi12d 473	. . 3 |- ( y = A -> ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
8		sban 2008	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
9	4, 7, 8	vtoclbg 2865	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
10	1, 3, 9	pm5.21nii 711	1 |- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   [wsb 1810   e. wcel 2202   _Vcvv 2802   [.wsbc 3031

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032

This theorem is referenced by:  sbc3an  3093  difopab  4863  sbcfung  5350  sbcfng  5480  sbcfg  5481  f1od2  6399