Theorem hbeleqOLD 1998

Description: If x is effectively bound in y e. A, then it is effectively bound in y = A.

Hypothesis

Ref	Expression
hbeleq.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbeleqOLD	|- (y = A -> A.x y = A)

Distinct variable groups:   x,y   y,A

Proof of Theorem hbeleqOLD

Step	Hyp	Ref	Expression
1		ax-17 1317	. 2 |- (z e. y -> A.x z e. y)
2		ax-17 1317	. . 3 |- (y e. z -> A.x y e. z)
3		hbeleq.1	. . 3 |- (y e. A -> A.x y e. A)
4	2, 3	hbel 1996	. 2 |- (z e. A -> A.x z e. A)
5	1, 4	hbeq 1995	1 |- (y = A -> A.x y = A)

Syntax hints:   -> wi 3  A.wal 1296   = wceq 1298   e. wcel 1300

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-cleq 1877  df-clel 1880

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