Theorem csbcnv 5306

Description: Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5307 without a sethood antecedent but depending on more axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbcnv	|- `' [_ A / x ]_ F = [_ A / x ]_ `' F

Proof of Theorem csbcnv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcbr 4707	. . . 4 |- ( [. A / x ]. z F y <-> z [_ A / x ]_ F y )
2	1	opabbii 4717	. . 3 |- { <. y , z >. | [. A / x ]. z F y } = { <. y , z >. | z [_ A / x ]_ F y }
3		csbopab 5008	. . 3 |- [_ A / x ]_ { <. y , z >. | z F y } = { <. y , z >. | [. A / x ]. z F y }
4		df-cnv 5122	. . 3 |- `' [_ A / x ]_ F = { <. y , z >. | z [_ A / x ]_ F y }
5	2, 3, 4	3eqtr4ri 2655	. 2 |- `' [_ A / x ]_ F = [_ A / x ]_ { <. y , z >. | z F y }
6		df-cnv 5122	. . 3 |- `' F = { <. y , z >. | z F y }
7	6	csbeq2i 3993	. 2 |- [_ A / x ]_ `' F = [_ A / x ]_ { <. y , z >. | z F y }
8	5, 7	eqtr4i 2647	1 |- `' [_ A / x ]_ F = [_ A / x ]_ `' F

Syntax hints:   = wceq 1483   [.wsbc 3435   [_csb 3533   class class class wbr 4653   {copab 4712   `'ccnv 5113

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-cnv 5122

This theorem is referenced by:  esum2dlem  30154  csbpredg  33172