Theorem sbc3an 3047

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 982	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2	1	sbcbii 3045	. . 3 |- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> [. A / x ]. ( ( ph /\ ps ) /\ ch ) )
3		sbcan 3028	. . 3 |- ( [. A / x ]. ( ( ph /\ ps ) /\ ch ) <-> ( [. A / x ]. ( ph /\ ps ) /\ [. A / x ]. ch ) )
4		sbcan 3028	. . . 4 |- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )
5	4	anbi1i 458	. . 3 |- ( ( [. A / x ]. ( ph /\ ps ) /\ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
6	2, 3, 5	3bitri 206	. 2 |- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
7		df-3an 982	. 2 |- ( ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
8	6, 7	bitr4i 187	1 |- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   [.wsbc 2985

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986

This theorem is referenced by: (None)