Theorem elab3 2412

Description: Membership in a class abstraction using implicit substitution.

Hypotheses

Ref	Expression
elab3.1	|- (ps -> A e. _V)
elab3.2	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
elab3	|- (A e. {x | ph} <-> ps)

Distinct variable groups:   ps,x   x,A

Proof of Theorem elab3

Step	Hyp	Ref	Expression
1		elab3.1	. 2 |- (ps -> A e. _V)
2		elab3.2	. . 3 |- (x = A -> (ph <-> ps))
3	2	elab3g 2409	. 2 |- ((ps -> A e. _V) -> (A e. {x | ph} <-> ps))
4	1, 3	ax-mp 7	1 |- (A e. {x | ph} <-> ps)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292

This theorem is referenced by:  fvelrnb 4719  oprvelrn 4969  elpm 5395  isfi 5441  elq 7437  eltg3 8896  islp 9020  islno 9753  elpi1 16089  isrnghom 16119  elstr 16714  iscsubsp 17209  isline 17220  ispoint 17223  ispsubsp 17226  ispsubcl 17347  ispaut 17396

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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