Theorem sbcan 3085

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 3051	. 2 |- ( [. A / x ]. ( ph /\ ps ) -> A e. _V )
2		sbcex 3051	. . 3 |- ( [. A / x ]. ps -> A e. _V )
3	2	adantl 277	. 2 |- ( ( [. A / x ]. ph /\ [. A / x ]. ps ) -> A e. _V )
4		dfsbcq2 3045	. . 3 |- ( y = A -> ( [ y / x ] ( ph /\ ps ) <-> [. A / x ]. ( ph /\ ps ) ) )
5		dfsbcq2 3045	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		dfsbcq2 3045	. . . 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 2009	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
9	4, 7, 8	vtoclbg 2876	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
10	1, 3, 9	pm5.21nii 712	1 |- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   [wsb 1811   e. wcel 2203   _Vcvv 2813   [.wsbc 3042

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-sbc 3043

This theorem is referenced by:  sbc3an  3104  difopab  4888  sbcfung  5376  sbcfng  5506  sbcfg  5507  f1od2  6431