Theorem sbcheg 38073

Description: Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)

Assertion

Ref	Expression
sbcheg	|- ( A e. V -> ( [. A / x ]. B hereditary C <-> [_ A / x ]_ B hereditary [_ A / x ]_ C ) )

Proof of Theorem sbcheg

Step	Hyp	Ref	Expression
1		sbcssg 4085	. . 3 |- ( A e. V -> ( [. A / x ]. ( B " C ) C_ C <-> [_ A / x ]_ ( B " C ) C_ [_ A / x ]_ C ) )
2		csbima12 5483	. . . . 5 |- [_ A / x ]_ ( B " C ) = ( [_ A / x ]_ B " [_ A / x ]_ C )
3	2	a1i 11	. . . 4 |- ( A e. V -> [_ A / x ]_ ( B " C ) = ( [_ A / x ]_ B " [_ A / x ]_ C ) )
4	3	sseq1d 3632	. . 3 |- ( A e. V -> ( [_ A / x ]_ ( B " C ) C_ [_ A / x ]_ C <-> ( [_ A / x ]_ B " [_ A / x ]_ C ) C_ [_ A / x ]_ C ) )
5	1, 4	bitrd 268	. 2 |- ( A e. V -> ( [. A / x ]. ( B " C ) C_ C <-> ( [_ A / x ]_ B " [_ A / x ]_ C ) C_ [_ A / x ]_ C ) )
6		df-he 38067	. . 3 |- ( B hereditary C <-> ( B " C ) C_ C )
7	6	sbcbii 3491	. 2 |- ( [. A / x ]. B hereditary C <-> [. A / x ]. ( B " C ) C_ C )
8		df-he 38067	. 2 |- ( [_ A / x ]_ B hereditary [_ A / x ]_ C <-> ( [_ A / x ]_ B " [_ A / x ]_ C ) C_ [_ A / x ]_ C )
9	5, 7, 8	3bitr4g 303	1 |- ( A e. V -> ( [. A / x ]. B hereditary C <-> [_ A / x ]_ B hereditary [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533   C_ wss 3574   "cima 5117   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  frege77  38234