Theorem vtocleg 1852

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 1816	. 2 |- (A e. B -> E.x x = A)
2		vtocleg.1	. . 3 |- (x = A -> ph)
3	2	19.23aiv 1294	. 2 |- (E.x x = A -> ph)
4	1, 3	syl 10	1 |- (A e. B -> ph)

Syntax hints:   -> wi 3   = wceq 955   e. wcel 957  E.wex 979

This theorem is referenced by:  vtocle 1855  a4sbc 1942  hbsbc1g 1945  ra4sbc 1994  noel 2281  prex 2777  avril1 8739

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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