Theorem abssi 2122

Description: Inference of abstraction subclass from implication.

Hypothesis

Ref	Expression
abssi.1	|- (ph -> x e. A)

Assertion

Ref	Expression
abssi	|- {x | ph} (_ A

Distinct variable group:   x,A

Proof of Theorem abssi

Step	Hyp	Ref	Expression
1		abssi.1	. . 3 |- (ph -> x e. A)
2	1	ss2abi 2120	. 2 |- {x | ph} (_ {x | x e. A}
3		abid2 1580	. 2 |- {x | x e. A} = A
4	2, 3	sseqtr 2093	1 |- {x | ph} (_ A

Syntax hints:   -> wi 3   e. wcel 958  {cab 1463   (_ wss 2047

This theorem is referenced by:  ssab2 2130  intab 2560  tfrlem8 3918  rankuni 4698  alephval3 4903  cfsuc 4915  limsupclt 6530  infcvgaux1 7219  tgval3t 7625  stcat 10457

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-in 2051  df-ss 2053

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