Theorem sbab 2578

Description: The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
sbab	|- ( x = y -> A = { z | [ y / x ] z e. A } )

Distinct variable groups:   z, A   x, z   y, z

Allowed substitution hints:   A( x, y)

Proof of Theorem sbab

Step	Hyp	Ref	Expression
1		sbequ12 2083	. 2 |- ( x = y -> ( z e. A <-> [ y / x ] z e. A ) )
2	1	abbi2dv 2570	1 |- ( x = y -> A = { z | [ y / x ] z e. A } )

Syntax hints:   -> wi 4   = wceq 1444   [wsb 1797   e. wcel 1887   {cab 2437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447

This theorem is referenced by:  sbcel12  3772  sbceqg  3773  sbcel12gOLD  36905