Theorem sbab 2604

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 1993	. 2 |- ( x = y -> ( z e. A <-> [ y / x ] z e. A ) )
2	1	abbi2dv 2594	1 |- ( x = y -> A = { z | [ y / x ] z e. A } )

Syntax hints:   -> wi 4   = wceq 1395   [wsb 1740   e. wcel 1819   {cab 2442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452

This theorem is referenced by:  sbcel12  3832  sbcel12gOLD  3833  sbceqg  3834