Theorem hbnel 2104

Description: Bound-variable hypothesis builder for inequality. (Contributed by by David Abernethy, 26-Jun-2011.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
hbnel.1	|- (y e. A -> A.x y e. A)
hbnel.2	|- (z e. B -> A.x z e. B)

Assertion

Ref	Expression
hbnel	|- (A e/ B -> A.x A e/ B)

Distinct variable groups:   y,A   z,B

Proof of Theorem hbnel

Step	Hyp	Ref	Expression
1		hbnel.1	. . . 4 |- (y e. A -> A.x y e. A)
2		hbnel.2	. . . 4 |- (z e. B -> A.x z e. B)
3	1, 2	hbel 1996	. . 3 |- (A e. B -> A.x A e. B)
4	3	hbn 1351	. 2 |- (-. A e. B -> A.x -. A e. B)
5		df-nel 2020	. 2 |- (A e/ B <-> -. A e. B)
6	5	albii 1346	. 2 |- (A.x A e/ B <-> A.x -. A e. B)
7	4, 5, 6	3imtr4i 236	1 |- (A e/ B -> A.x A e/ B)

Syntax hints:  -. wn 2   -> wi 3  A.wal 1296   e. wcel 1300   e/ wnel 2018

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-cleq 1877  df-clel 1880  df-nel 2020

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