Theorem hbpw 1804

Description: Bound-variable hypothesis builder for power class.

Hypothesis

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

Assertion

Ref	Expression
hbpw	|- (y e. P~A -> A.x y e. P~A)

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

Proof of Theorem hbpw

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 |- (z e. y -> A.x z e. y)
2		ax-17 925	. . . 4 |- (y e. z -> A.x y e. z)
3		hbpw.1	. . . 4 |- (y e. A -> A.x y e. A)
4	2, 3	hbel 1172	. . 3 |- (z e. A -> A.x z e. A)
5	1, 4	hbss 1501	. 2 |- (y (_ A -> A.x y (_ A)
6		visset 1350	. . 3 |- y e. V
7	6	elpw 1801	. 2 |- (y e. P~A <-> y (_ A)
8	7	bial 695	. 2 |- (A.x y e. P~A <-> A.x y (_ A)
9	5, 7, 8	3imtr4 192	1 |- (y e. P~A -> A.x y e. P~A)

Syntax hints:   -> wi 2  A.wal 672   e. wel 803   e. wcel 1092   (_ wss 1487  P~cpw 1798

This theorem is referenced by:  cardprc 3667

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-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799

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