Theorem hbn1w 1706

Description: Weak version of hbn1 1730. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
hbn1w.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
hbn1w	|- (-. A.xph -> A.x -. A.xph)

Distinct variable groups:   ph,y   ps,x   x,y

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

Proof of Theorem hbn1w

Step	Hyp	Ref	Expression
1		ax-17 1616	. 2 |- (A.xph -> A.yA.xph)
2		ax-17 1616	. 2 |- (-. ps -> A.x -. ps)
3		ax-17 1616	. 2 |- (A.yps -> A.xA.yps)
4		ax-17 1616	. 2 |- (-. ph -> A.y -. ph)
5		ax-17 1616	. 2 |- (-. A.yps -> A.x -. A.yps)
6		hbn1w.1	. 2 |- (x = y -> (ph <-> ps))
7	1, 2, 3, 4, 5, 6	hbn1fw 1705	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:  hba1w  1707  hbe1w  1708  ax6w  1717