Theorem hbn1fw 1705

Description: Weak version of ax-6 1729 from which we can prove any ax-6 1729 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Hypotheses

Ref	Expression
hbn1fw.1	|- (A.xph -> A.yA.xph)
hbn1fw.2	|- (-. ps -> A.x -. ps)
hbn1fw.3	|- (A.yps -> A.xA.yps)
hbn1fw.4	|- (-. ph -> A.y -. ph)
hbn1fw.5	|- (-. A.yps -> A.x -. A.yps)
hbn1fw.6	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
hbn1fw	|- (-. A.xph -> A.x -. A.xph)

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)   ps(x,y)

Proof of Theorem hbn1fw

Step	Hyp	Ref	Expression
1		hbn1fw.1	. . . . 5 |- (A.xph -> A.yA.xph)
2		hbn1fw.2	. . . . 5 |- (-. ps -> A.x -. ps)
3		hbn1fw.3	. . . . 5 |- (A.yps -> A.xA.yps)
4		hbn1fw.4	. . . . 5 |- (-. ph -> A.y -. ph)
5		hbn1fw.6	. . . . 5 |- (x = y -> (ph <-> ps))
6	1, 2, 3, 4, 5	cbvalw 1701	. . . 4 |- (A.xph <-> A.yps)
7	6	biimpri 197	. . 3 |- (A.yps -> A.xph)
8	7	con3i 127	. 2 |- (-. A.xph -> -. A.yps)
9		hbn1fw.5	. 2 |- (-. A.yps -> A.x -. A.yps)
10	6	biimpi 186	. . . 4 |- (A.xph -> A.yps)
11	10	con3i 127	. . 3 |- (-. A.yps -> -. A.xph)
12	11	alimi 1559	. 2 |- (A.x -. A.yps -> A.x -. A.xph)
13	8, 9, 12	3syl 18	1 |- (-. A.xph -> A.x -. A.xph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  hbn1w  1706