Theorem rabeq 1809

Description: Equality theorem for restricted class abstractions.

Assertion

Ref	Expression
rabeq	|- (A = B -> {x e. A | ph} = {x e. B | ph})

Distinct variable groups:   x,A   x,B

Proof of Theorem rabeq

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (y e. A -> A.x y e. A)
2		ax-17 971	. 2 |- (y e. B -> A.x y e. B)
3	1, 2	rabeqf 1808	1 |- (A = B -> {x e. A | ph} = {x e. B | ph})

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  {crab 1648

This theorem is referenced by:  scott0 4717  acdc3lem 7486  acdc3 7487  acdc2lem1 7488  acdc2 7490  acdc5lem1 7491  acdc5 7493  acdclem 7494  acdc 7495  ntrfval 7667  clsfval 7668  cnfval 7756  cnpfval 7757  blfval 7835  grpidval 8058  grpinvfval 8066  bloval 8441  hmoval 8470  spwval2 8653  spwval 8659  hhblo 9828  elgiso 10398  sfvlim 10605  sfvlimOLD 10606  limfillem2OLD 10608

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  df-rab 1652

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