Theorem hblem 1170

Description: Lemma for hbeq 1171 and hbel 1172.

Hypothesis

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

Assertion

Ref	Expression
hblem	|- (z e. A -> A.x z e. A)

Distinct variable group(s):   y,A   x,y,z

Proof of Theorem hblem

Step	Hyp	Ref	Expression
1		eleq1 1149	. . 3 |- (y = z -> (y e. A <-> z e. A))
2	1	bialdv 935	. . 3 |- (y = z -> (A.x y e. A <-> A.x z e. A))
3	1, 2	imbi12d 474	. 2 |- (y = z -> ((y e. A -> A.x y e. A) <-> (z e. A -> A.x z e. A)))
4		hblem.1	. 2 |- (y e. A -> A.x y e. A)
5	3, 4	chv 984	1 |- (z e. A -> A.x z e. A)

Syntax hints:   -> wi 2  A.wal 672   = weq 797   e. wcel 1092

This theorem is referenced by:  hbeq 1171  hbel 1172

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099

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