Theorem sbab 2305

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 1771	. 2 |- ( x = y -> ( z e. A <-> [ y / x ] z e. A ) )
2	1	abbi2dv 2296	1 |- ( x = y -> A = { z | [ y / x ] z e. A } )

Syntax hints:   -> wi 4   = wceq 1353   [wsb 1762   e. wcel 2148   {cab 2163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173

This theorem is referenced by:  sbcel12g  3072  sbceqg  3073