Theorem hbe1w 1976

Description: Weak version of hbe1 2021. See comments for ax10w 2006. 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. x ph -> A. x E. x ph )

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 1705	. 2 |- ( E. x ph <-> -. A. x -. ph )
2		hbn1w.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	notbid 308	. . 3 |- ( x = y -> ( -. ph <-> -. ps ) )
4	3	hbn1w 1973	. 2 |- ( -. A. x -. ph -> A. x -. A. x -. ph )
5	1, 4	hbxfrbi 1752	1 |- ( E. x ph -> A. x E. x ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)