Theorem sbab 2164

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 1654	. 2 |- ( x = y -> ( z e. A <-> [ y / x ] z e. A ) )
2	1	abbi2dv 2156	1 |- ( x = y -> A = { z | [ y / x ] z e. A } )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   [wsb 1645   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036

This theorem is referenced by:  sbcel12g  2865  sbceqg  2866