Theorem sbab 1583

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 1181	. 2 |- (x = y -> (z e. A <-> [y / x]z e. A))
2	1	abbi2dv 1578	1 |- (x = y -> A = {z | [y / x]z e. A})

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463

This theorem is referenced by:  moop2 2801  fvopabgf 3787  fvopabnf 3788  oprabval4g 4031  seq1lem1 6309  fsum1f 7007  fsump1f 7011

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472

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