Theorem hbpw 2407

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 groups:   y,A   x,y

Proof of Theorem hbpw

Step	Hyp	Ref	Expression
1		ax-17 971	. . 3 |- (z e. y -> A.x z e. y)
2		ax-17 971	. . . 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 1566	. . 3 |- (z e. A -> A.x z e. A)
5	1, 4	hbss 2062	. 2 |- (y (_ A -> A.x y (_ A)
6		visset 1813	. . 3 |- y e. V
7	6	elpw 2404	. 2 |- (y e. P~A <-> y (_ A)
8	7	albii 999	. 2 |- (A.x y e. P~A <-> A.x y (_ A)
9	5, 7, 8	3imtr4 219	1 |- (y e. P~A -> A.x y e. P~A)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  cardprc 4861  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402

Copyright terms: Public domain