Theorem abssdv 2124

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 1288	. 2 |- (ph -> A.x(ps -> x e. A))
3		abss 2120	. 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 956   e. wcel 960  {cab 1466   (_ wss 2050

This theorem is referenced by:  lpsscls 7742  nmosetre 8423  nmopsetretALT 9785  nmfnsetret 9799

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056

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