Theorem csbcnv 5191

Description: Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5192 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

Proof of Theorem csbcnv

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

Step	Hyp	Ref	Expression
1		sbcbr 4505	. . . 4
2	1	opabbii 4516	. . 3
3		csbopab 4784	. . 3
4		df-cnv 5012	. . 3
5	2, 3, 4	3eqtr4ri 2497	. 2
6		df-cnv 5012	. . 3
7	6	csbeq2i 3836	. 2
8	5, 7	eqtr4i 2489	1

Syntax hints:  =wceq 1395  [.wsbc 3327  [_csb 3434   class class class wbr 4452  {copab 4509  `'ccnv 5003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-cnv 5012