Theorem hbsn 1833

Description: Bound-variable hypothesis builder for singletons.

Hypothesis

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

Assertion

Ref	Expression
hbsn	|- (y e. {A} -> A.x y e. {A})

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

Proof of Theorem hbsn

Step	Hyp	Ref	Expression
1		hbsn.1	. . 3 |- (y e. A -> A.x y e. A)
2	1, 1	hbpr 1824	. 2 |- (y e. {A, A} -> A.x y e. {A, A})
3		dfsn2 1819	. . 3 |- {A} = {A, A}
4	3	eleq2i 1153	. 2 |- (y e. {A} <-> y e. {A, A})
5	4	bial 695	. 2 |- (A.x y e. {A} <-> A.x y e. {A, A})
6	2, 4, 5	3imtr4 192	1 |- (y e. {A} -> A.x y e. {A})

Syntax hints:   -> wi 2  A.wal 672   e. wcel 1092  {csn 1808  {cpr 1809

This theorem is referenced by:  hbop 1879  hbfv 2837

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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