Theorem hbcnv 3301

Description: Bound-variable hypothesis builder for converse.

Hypothesis

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

Assertion

Ref	Expression
hbcnv	|- (y e. `'A -> A.x y e. `'A)

Distinct variable groups:   y,A   x,y

Proof of Theorem hbcnv

Step	Hyp	Ref	Expression
1		ax-17 973	. . . . 5 |- (y = <.z, w>. -> A.x y = <.z, w>.)
2		ax-17 973	. . . . . 6 |- (y e. w -> A.x y e. w)
3		hbcnv.1	. . . . . 6 |- (y e. A -> A.x y e. A)
4		ax-17 973	. . . . . 6 |- (y e. z -> A.x y e. z)
5	2, 3, 4	hbbr 2663	. . . . 5 |- (wAz -> A.x wAz)
6	1, 5	hban 1011	. . . 4 |- ((y = <.z, w>. /\ wAz) -> A.x(y = <.z, w>. /\ wAz))
7	6	hbex 1008	. . 3 |- (E.w(y = <.z, w>. /\ wAz) -> A.xE.w(y = <.z, w>. /\ wAz))
8	7	hbex 1008	. 2 |- (E.zE.w(y = <.z, w>. /\ wAz) -> A.xE.zE.w(y = <.z, w>. /\ wAz))
9		elcnv 3299	. 2 |- (y e. `'A <-> E.zE.w(y = <.z, w>. /\ wAz))
10	9	albii 1001	. 2 |- (A.x y e. `'A <-> A.xE.zE.w(y = <.z, w>. /\ wAz))
11	8, 9, 10	3imtr4 219	1 |- (y e. `'A -> A.x y e. `'A)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  <.cop 2415   class class class wbr 2624  `'ccnv 3175

This theorem is referenced by:  hbdm 3358  hbfun 3542  hbf1 3669  cnvtr 10609

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-cnv 3192

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