Theorem hbe1w 1708

Description: Weak version of hbe1 1731. See comments for ax6w 1717. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Hypothesis

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

Assertion

Ref	Expression
hbe1w	|- (E.xph -> A.xE.xph)

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

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

Proof of Theorem hbe1w

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 |- (E.xph <-> -. A.x -. ph)
2		hbn1w.1	. . . 4 |- (x = y -> (ph <-> ps))
3	2	notbid 285	. . 3 |- (x = y -> (-. ph <-> -. ps))
4	3	hbn1w 1706	. 2 |- (-. A.x -. ph -> A.x -. A.x -. ph)
5	1, 4	hbxfrbi 1568	1 |- (E.xph -> A.xE.xph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541

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: (None)