Theorem csbeq2 3064

Description: Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Assertion

Ref	Expression
csbeq2	|- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C )

Proof of Theorem csbeq2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2228	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	alimi 1442	. . . 4 |- ( A. x B = C -> A. x ( y e. B <-> y e. C ) )
3		sbcbi2 2996	. . . 4 |- ( A. x ( y e. B <-> y e. C ) -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
4	2, 3	syl 14	. . 3 |- ( A. x B = C -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
5	4	abbidv 2282	. 2 |- ( A. x B = C -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 3041	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 3041	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2222	1 |- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1340   = wceq 1342   e. wcel 2135   {cab 2150   [.wsbc 2946   [_csb 3040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-sbc 2947  df-csb 3041

This theorem is referenced by:  prodeq2w  11483