Theorem sbcrel 4856

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 3633	. . 3 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) ) )
2		csbconstg 3161	. . . 4 |- ( A e. V -> [_ A / x ]_ ( _V X. _V ) = ( _V X. _V ) )
3	2	sseq2d 3278	. . 3 |- ( A e. V -> ( [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
4	1, 3	bitrd 188	. 2 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
5		df-rel 4776	. . 3 |- ( Rel R <-> R C_ ( _V X. _V ) )
6	5	sbcbii 3111	. 2 |- ( [. A / x ]. Rel R <-> [. A / x ]. R C_ ( _V X. _V ) )
7		df-rel 4776	. 2 |- ( Rel [_ A / x ]_ R <-> [_ A / x ]_ R C_ ( _V X. _V ) )
8	4, 6, 7	3bitr4g 223	1 |- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2209   _Vcvv 2821   [.wsbc 3051   [_csb 3147   C_ wss 3220   X. cxp 4767   Rel wrel 4774

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  df-csb 3148  df-in 3226  df-ss 3233  df-rel 4776

This theorem is referenced by:  sbcfung  5396