Theorem hbex 1689

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 1548	. . 3 |- ( E. y ph -> A. y E. y ph )
2	1	hbal 1530	. 2 |- ( A. x E. y ph -> A. y A. x E. y ph )
3		hbex.1	. . 3 |- ( ph -> A. x ph )
4		19.8a 1643	. . 3 |- ( ph -> E. y ph )
5	3, 4	alrimih 1522	. 2 |- ( ph -> A. x E. y ph )
6	2, 5	exlimih 1646	1 |- ( E. y ph -> A. x E. y ph )

Syntax hints:   -> wi 4   A.wal 1400   E.wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfex  1690  excomim  1715  19.12  1717  cbvexh  1808  cbvexdh  1982  hbsbv  2001  hbeu1  2096  hbmo  2125  moexexdc  2171