Theorem hbex 1527

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 1384	. . 3 |- ( E. y ph -> A. y E. y ph )
2	1	hbal 1366	. 2 |- ( A. x E. y ph -> A. y A. x E. y ph )
3		hbex.1	. . 3 |- ( ph -> A. x ph )
4		19.8a 1482	. . 3 |- ( ph -> E. y ph )
5	3, 4	alrimih 1358	. 2 |- ( ph -> A. x E. y ph )
6	2, 5	exlimih 1484	1 |- ( E. y ph -> A. x E. y ph )

Syntax hints:   -> wi 4   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  nfex  1528  excomim  1553  19.12  1555  cbvexh  1638  cbvexdh  1801  hbsbv  1817  hbeu1  1910  hbmo  1939  moexexdc  1984