Theorem sbc3ang 2820

Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbc3ang	|- ( A e. V -> ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) ) )

Proof of Theorem sbc3ang

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2767	. 2 |- ( y = A -> ( [ y / x ] ( ph /\ ps /\ ch ) <-> [. A / x ]. ( ph /\ ps /\ ch ) ) )
2		dfsbcq2 2767	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 2767	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4		dfsbcq2 2767	. . 3 |- ( y = A -> ( [ y / x ] ch <-> [. A / x ]. ch ) )
5	2, 3, 4	3anbi123d 1207	. 2 |- ( y = A -> ( ( [ y / x ] ph /\ [ y / x ] ps /\ [ y / x ] ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) ) )
6		sb3an 1832	. 2 |- ( [ y / x ] ( ph /\ ps /\ ch ) <-> ( [ y / x ] ph /\ [ y / x ] ps /\ [ y / x ] ch ) )
7	1, 5, 6	vtoclbg 2614	1 |- ( A e. V -> ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) ) )

Syntax hints:   -> wi 4   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   [wsb 1645   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by: (None)