Theorem sbcan 2975

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 2941	. 2 |- ( [. A / x ]. ( ph /\ ps ) -> A e. _V )
2		sbcex 2941	. . 3 |- ( [. A / x ]. ps -> A e. _V )
3	2	adantl 275	. 2 |- ( ( [. A / x ]. ph /\ [. A / x ]. ps ) -> A e. _V )
4		dfsbcq2 2936	. . 3 |- ( y = A -> ( [ y / x ] ( ph /\ ps ) <-> [. A / x ]. ( ph /\ ps ) ) )
5		dfsbcq2 2936	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		dfsbcq2 2936	. . . 4 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
7	5, 6	anbi12d 465	. . 3 |- ( y = A -> ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
8		sban 1932	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
9	4, 7, 8	vtoclbg 2770	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
10	1, 3, 9	pm5.21nii 694	1 |- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   [wsb 1739   e. wcel 2125   _Vcvv 2709   [.wsbc 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-sbc 2934

This theorem is referenced by:  sbc3an  2994  difopab  4712  sbcfung  5187  sbcfng  5310  sbcfg  5311  f1od2  6172