Theorem hbex 1567

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 1424	. . 3 |- ( E. y ph -> A. y E. y ph )
2	1	hbal 1406	. 2 |- ( A. x E. y ph -> A. y A. x E. y ph )
3		hbex.1	. . 3 |- ( ph -> A. x ph )
4		19.8a 1522	. . 3 |- ( ph -> E. y ph )
5	3, 4	alrimih 1398	. 2 |- ( ph -> A. x E. y ph )
6	2, 5	exlimih 1524	1 |- ( E. y ph -> A. x E. y ph )

Syntax hints:   -> wi 4   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nfex  1568  excomim  1593  19.12  1595  cbvexh  1678  cbvexdh  1842  hbsbv  1858  hbeu1  1951  hbmo  1980  moexexdc  2025