Theorem csbeq2 3069

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 2230	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	alimi 1443	. . . 4 |- ( A. x B = C -> A. x ( y e. B <-> y e. C ) )
3		sbcbi2 3001	. . . 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 2284	. 2 |- ( A. x B = C -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 3046	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 3046	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2224	1 |- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   {cab 2151   [.wsbc 2951   [_csb 3045

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952  df-csb 3046

This theorem is referenced by:  prodeq2w  11497