Theorem csbcnvgALT 5307

Description: Move class substitution in and out of the converse of a function. Version of csbcnv 5306 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbcnvgALT	|- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )

Proof of Theorem csbcnvgALT

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

Step	Hyp	Ref	Expression
1		sbcbr123 4706	. . . . 5 |- ( [. A / x ]. z F y <-> [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y )
2		csbconstg 3546	. . . . . 6 |- ( A e. V -> [_ A / x ]_ z = z )
3		csbconstg 3546	. . . . . 6 |- ( A e. V -> [_ A / x ]_ y = y )
4	2, 3	breq12d 4666	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y <-> z [_ A / x ]_ F y ) )
5	1, 4	syl5bb 272	. . . 4 |- ( A e. V -> ( [. A / x ]. z F y <-> z [_ A / x ]_ F y ) )
6	5	opabbidv 4716	. . 3 |- ( A e. V -> { <. y , z >. | [. A / x ]. z F y } = { <. y , z >. | z [_ A / x ]_ F y } )
7		csbopabgALT 5009	. . 3 |- ( A e. V -> [_ A / x ]_ { <. y , z >. | z F y } = { <. y , z >. | [. A / x ]. z F y } )
8		df-cnv 5122	. . . 4 |- `' [_ A / x ]_ F = { <. y , z >. | z [_ A / x ]_ F y }
9	8	a1i 11	. . 3 |- ( A e. V -> `' [_ A / x ]_ F = { <. y , z >. | z [_ A / x ]_ F y } )
10	6, 7, 9	3eqtr4rd 2667	. 2 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ { <. y , z >. | z F y } )
11		df-cnv 5122	. . 3 |- `' F = { <. y , z >. | z F y }
12	11	csbeq2i 3993	. 2 |- [_ A / x ]_ `' F = [_ A / x ]_ { <. y , z >. | z F y }
13	10, 12	syl6eqr 2674	1 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [.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

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-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: (None)