Theorem hbne 1641

Description: Bound-variable hypothesis builder for inequality.

Hypotheses

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

Assertion

Ref	Expression
hbne	|- (A =/= B -> A.x A =/= B)

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem hbne

Step	Hyp	Ref	Expression
1		hbne.1	. . . 4 |- (y e. A -> A.x y e. A)
2		hbne.2	. . . 4 |- (y e. B -> A.x y e. B)
3	1, 2	hbeq 1562	. . 3 |- (A = B -> A.x A = B)
4	3	hbn 1002	. 2 |- (-. A = B -> A.x -. A = B)
5		df-ne 1584	. 2 |- (A =/= B <-> -. A = B)
6	5	albii 997	. 2 |- (A.x A =/= B <-> A.x -. A = B)
7	4, 5, 6	3imtr4 219	1 |- (A =/= B -> A.x A =/= B)

Syntax hints:  -. wn 2   -> wi 3  A.wal 952   = wceq 954   e. wcel 956   =/= wne 1582

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-cleq 1467  df-clel 1470  df-ne 1584

Copyright terms: Public domain