Theorem sbcan 3032

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 2998	. 2 |- ( [. A / x ]. ( ph /\ ps ) -> A e. _V )
2		sbcex 2998	. . 3 |- ( [. A / x ]. ps -> A e. _V )
3	2	adantl 277	. 2 |- ( ( [. A / x ]. ph /\ [. A / x ]. ps ) -> A e. _V )
4		dfsbcq2 2992	. . 3 |- ( y = A -> ( [ y / x ] ( ph /\ ps ) <-> [. A / x ]. ( ph /\ ps ) ) )
5		dfsbcq2 2992	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		dfsbcq2 2992	. . . 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 1974	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
9	4, 7, 8	vtoclbg 2825	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
10	1, 3, 9	pm5.21nii 705	1 |- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   [wsb 1776   e. wcel 2167   _Vcvv 2763   [.wsbc 2989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  sbc3an  3051  difopab  4799  sbcfung  5282  sbcfng  5405  sbcfg  5406  f1od2  6293