Theorem vtocleg 1855

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 1819	. 2 |- (A e. B -> E.x x = A)
2		vtocleg.1	. . 3 |- (x = A -> ph)
3	2	19.23aiv 1295	. 2 |- (E.x x = A -> ph)
4	1, 3	syl 10	1 |- (A e. B -> ph)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  vtocle 1858  a4sbc 1945  hbsbc1g 1948  ra4sbc 1997  noel 2284  prex 2781  avril1 8784

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  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-v 1812

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