Theorem csbeq2 3537

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 2690	. . . . 5 |- ( B = C -> ( y e. B <-> y e. C ) )
2	1	alimi 1739	. . . 4 |- ( A. x B = C -> A. x ( y e. B <-> y e. C ) )
3		sbcbi2 3484	. . . 4 |- ( A. x ( y e. B <-> y e. C ) -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
4	2, 3	syl 17	. . 3 |- ( A. x B = C -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
5	4	abbidv 2741	. 2 |- ( A. x B = C -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 3534	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 3534	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2681	1 |- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436  df-csb 3534

This theorem is referenced by:  sumeq2w  14422  prodeq2w  14642  csbeq12  33966  csbfv12gALTVD  39135