Theorem sbcan 3094

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 3060	. 2 |- ( [. A / x ]. ( ph /\ ps ) -> A e. _V )
2		sbcex 3060	. . 3 |- ( [. A / x ]. ps -> A e. _V )
3	2	adantl 277	. 2 |- ( ( [. A / x ]. ph /\ [. A / x ]. ps ) -> A e. _V )
4		dfsbcq2 3054	. . 3 |- ( y = A -> ( [ y / x ] ( ph /\ ps ) <-> [. A / x ]. ( ph /\ ps ) ) )
5		dfsbcq2 3054	. . . 4 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		dfsbcq2 3054	. . . 4 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
7	5, 6	anbi12d 477	. . 3 |- ( y = A -> ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
8		sban 2015	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
9	4, 7, 8	vtoclbg 2884	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) ) )
10	1, 3, 9	pm5.21nii 716	1 |- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   = 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:  sbc3an  3113  difopab  4908  sbcfung  5396  sbcfng  5526  sbcfg  5527  f1od2  6461