Theorem hbex 2156

Description: If x is not free in ph, it is not free in E. y ph. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2154, hbex 2156. (Revised by Wolf Lammen, 16-Oct-2021.)

Hypothesis

Ref	Expression
hbex.1	|- ( ph -> A. x ph )

Assertion

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

Proof of Theorem hbex

Step	Hyp	Ref	Expression
1		hbex.1	. . . 4 |- ( ph -> A. x ph )
2	1	nf5i 2024	. . 3 |- F/ x ph
3	2	nfex 2154	. 2 |- F/ x E. y ph
4	3	nf5ri 2065	1 |- ( E. y ph -> A. x E. y ph )

Syntax hints:   -> 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-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)