Theorem hbex 1684

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 1543	. . 3 |- ( E. y ph -> A. y E. y ph )
2	1	hbal 1525	. 2 |- ( A. x E. y ph -> A. y A. x E. y ph )
3		hbex.1	. . 3 |- ( ph -> A. x ph )
4		19.8a 1638	. . 3 |- ( ph -> E. y ph )
5	3, 4	alrimih 1517	. 2 |- ( ph -> A. x E. y ph )
6	2, 5	exlimih 1641	1 |- ( E. y ph -> A. x E. y ph )

Syntax hints:   -> wi 4   A.wal 1395   E.wex 1540

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfex  1685  excomim  1711  19.12  1713  cbvexh  1803  cbvexdh  1975  hbsbv  1994  hbeu1  2089  hbmo  2118  moexexdc  2164