Theorem sbab 1575

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 1177	. 2 |- (x = y -> (z e. A <-> [y / x]z e. A))
2	1	abbi2dv 1570	1 |- (x = y -> A = {z | [y / x]z e. A})

Syntax hints:   -> wi 3   = wceq 953   e. wcel 955  [wsbc 1166  {cab 1456

This theorem is referenced by:  moop2 2790  fvopabgf 3772  fvopabnf 3773  oprabval4g 4016  seq1lem1 6246  fsum1f 6945  fsump1f 6949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465

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