Theorem hbex 1568

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 1425	. . 3 |- ( E. y ph -> A. y E. y ph )
2	1	hbal 1407	. 2 |- ( A. x E. y ph -> A. y A. x E. y ph )
3		hbex.1	. . 3 |- ( ph -> A. x ph )
4		19.8a 1523	. . 3 |- ( ph -> E. y ph )
5	3, 4	alrimih 1399	. 2 |- ( ph -> A. x E. y ph )
6	2, 5	exlimih 1525	1 |- ( E. y ph -> A. x E. y ph )

Syntax hints:   -> wi 4   A.wal 1283   E.wex 1422

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nfex  1569  excomim  1594  19.12  1596  cbvexh  1680  cbvexdh  1844  hbsbv  1860  hbeu1  1953  hbmo  1982  moexexdc  2027