Theorem csbeq2 3104

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 2257	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	alimi 1466	. . . 4 |- ( A. x B = C -> A. x ( y e. B <-> y e. C ) )
3		sbcbi2 3036	. . . 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 2311	. 2 |- ( A. x B = C -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 3081	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 3081	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2251	1 |- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2164   {cab 2179   [.wsbc 2985   [_csb 3080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2986  df-csb 3081

This theorem is referenced by:  prodeq2w  11699