Theorem hbe1a 2022

Description: Dual statement of hbe1 2021. Modified version of axc7e 2133 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)

Assertion

Ref	Expression
hbe1a	|- ( E. x A. x ph -> A. x ph )

Proof of Theorem hbe1a

Step	Hyp	Ref	Expression
1		df-ex 1705	. 2 |- ( E. x A. x ph <-> -. A. x -. A. x ph )
2		hbn1 2020	. . 3 |- ( -. A. x ph -> A. x -. A. x ph )
3	2	con1i 144	. 2 |- ( -. A. x -. A. x ph -> A. x ph )
4	1, 3	sylbi 207	1 |- ( E. x A. x ph -> A. x ph )

Syntax hints:   -. wn 3   -> wi 4   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-10 2019

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

This theorem is referenced by:  nf5-1  2023  axc7e  2133  wl-dveeq12  33311