Theorem abssi 2118

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 2116	. 2 |- {x | ph} (_ {x | x e. A}
3		abid2 1577	. 2 |- {x | x e. A} = A
4	2, 3	sseqtr 2089	1 |- {x | ph} (_ A

Syntax hints:   -> wi 3   e. wcel 956  {cab 1461   (_ wss 2043

This theorem is referenced by:  ssab2 2126  intab 2555  tfrlem8 3909  rankuni 4678  alephval3 4883  cfsuc 4895  limsupclt 6470  infcvgaux1 7162  tgval3t 7575  stcat 10389

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-in 2047  df-ss 2049

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