Theorem sbab 2265

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 1744	. 2 |- ( x = y -> ( z e. A <-> [ y / x ] z e. A ) )
2	1	abbi2dv 2256	1 |- ( x = y -> A = { z | [ y / x ] z e. A } )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   [wsb 1735   {cab 2123

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133

This theorem is referenced by:  sbcel12g  3012  sbceqg  3013