Theorem vtocleg 1858

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 1822	. 2 |- (A e. B -> E.x x = A)
2		vtocleg.1	. . 3 |- (x = A -> ph)
3	2	19.23aiv 1297	. 2 |- (E.x x = A -> ph)
4	1, 3	syl 10	1 |- (A e. B -> ph)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  E.wex 982

This theorem is referenced by:  vtocle 1861  a4sbc 1948  hbsbc1g 1951  ra4sbc 2000  noel 2287  prex 2787  avril1 8779

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  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-v 1815

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