Theorem sbc3an 2998

Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbc3an	|- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) )

Proof of Theorem sbc3an

Step	Hyp	Ref	Expression
1		df-3an 965	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2	1	sbcbii 2996	. . 3 |- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> [. A / x ]. ( ( ph /\ ps ) /\ ch ) )
3		sbcan 2979	. . 3 |- ( [. A / x ]. ( ( ph /\ ps ) /\ ch ) <-> ( [. A / x ]. ( ph /\ ps ) /\ [. A / x ]. ch ) )
4		sbcan 2979	. . . 4 |- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )
5	4	anbi1i 454	. . 3 |- ( ( [. A / x ]. ( ph /\ ps ) /\ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
6	2, 3, 5	3bitri 205	. 2 |- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
7		df-3an 965	. 2 |- ( ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
8	6, 7	bitr4i 186	1 |- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 963   [.wsbc 2937

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938

This theorem is referenced by: (None)