Theorem vtocleg 2355

Description: Implicit substitution of a class for a set variable.

Hypothesis

Ref	Expression
vtocleg.1	|- (x = A -> ph)

Assertion

Ref	Expression
vtocleg	|- (A e. B -> ph)

Distinct variable groups:   x,A   ph,x

Proof of Theorem vtocleg

Step	Hyp	Ref	Expression
1		elex 2302	. 2 |- (A e. B -> E.x x = A)
2		vtocleg.1	. . 3 |- (x = A -> ph)
3	2	19.23aiv 1674	. 2 |- (E.x x = A -> ph)
4	1, 3	syl 12	1 |- (A e. B -> ph)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem is referenced by:  vtocle 2359  a4sbc 2457  hbsbc1g 2461  ra4sbc 2536  noel 2879  prex 3526  avril1 10142

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  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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