Theorem abssdv 2121

Description: Deduction of abstraction subclass from implication.

Hypothesis

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

Assertion

Ref	Expression
abssdv	|- (ph -> {x | ps} (_ A)

Distinct variable groups:   ph,x   x,A

Proof of Theorem abssdv

Step	Hyp	Ref	Expression
1		abssdv.1	. . 3 |- (ph -> (ps -> x e. A))
2	1	19.21aiv 1286	. 2 |- (ph -> A.x(ps -> x e. A))
3		abss 2117	. 2 |- ({x | ps} (_ A <-> A.x(ps -> x e. A))
4	2, 3	sylibr 200	1 |- (ph -> {x | ps} (_ A)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  {cab 1463   (_ wss 2047

This theorem is referenced by:  lpsscls 7745  nmosetre 8427  nmopsetretALT 9790  nmfnsetret 9804

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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