Theorem hbex 1616

Description: If x is not free in ph, it is not free in E. y ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

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		hbe1 1475	. . 3 |- ( E. y ph -> A. y E. y ph )
2	1	hbal 1457	. 2 |- ( A. x E. y ph -> A. y A. x E. y ph )
3		hbex.1	. . 3 |- ( ph -> A. x ph )
4		19.8a 1570	. . 3 |- ( ph -> E. y ph )
5	3, 4	alrimih 1449	. 2 |- ( ph -> A. x E. y ph )
6	2, 5	exlimih 1573	1 |- ( E. y ph -> A. x E. y ph )

Syntax hints:   -> wi 4   A.wal 1333   E.wex 1472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nfex  1617  excomim  1643  19.12  1645  cbvexh  1735  cbvexdh  1906  hbsbv  1921  hbeu1  2016  hbmo  2045  moexexdc  2090