Theorem sbcrel 4426

Description: Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Assertion

Ref	Expression
sbcrel	|- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) )

Proof of Theorem sbcrel

Step	Hyp	Ref	Expression
1		sbcssg 3330	. . 3 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) ) )
2		csbconstg 2864	. . . 4 |- ( A e. V -> [_ A / x ]_ ( _V X. _V ) = ( _V X. _V ) )
3	2	sseq2d 2973	. . 3 |- ( A e. V -> ( [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
4	1, 3	bitrd 177	. 2 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
5		df-rel 4352	. . 3 |- ( Rel R <-> R C_ ( _V X. _V ) )
6	5	sbcbii 2818	. 2 |- ( [. A / x ]. Rel R <-> [. A / x ]. R C_ ( _V X. _V ) )
7		df-rel 4352	. 2 |- ( Rel [_ A / x ]_ R <-> [_ A / x ]_ R C_ ( _V X. _V ) )
8	4, 6, 7	3bitr4g 212	1 |- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   _Vcvv 2557   [.wsbc 2764   [_csb 2852   C_ wss 2917   X. cxp 4343   Rel wrel 4350

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-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  df-csb 2853  df-in 2924  df-ss 2931  df-rel 4352

This theorem is referenced by:  sbcfung  4925