Theorem sbab 1590

Description: The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable.

Assertion

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

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

Proof of Theorem sbab

Step	Hyp	Ref	Expression
1		sbequ12 1185	. 2 |- (x = y -> (z e. A <-> [y / x]z e. A))
2	1	abbi2dv 1585	1 |- (x = y -> A = {z | [y / x]z e. A})

Syntax hints:   -> wi 3   = wceq 960   e. wcel 962  [wsbc 1174  {cab 1469

This theorem is referenced by:  moop2 2817  fvopabgf 3803  fvopabnf 3804  oprabval4g 4047  seq1lem1 6510  fsum1fi 7039  fsump1fi 7043

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478

Copyright terms: Public domain