Theorem hbex 1841

Description: If x is not free in ph, it is not free in E.yph. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbex.1	|- (ph -> A.xph)

Assertion

Ref	Expression
hbex	|- (E.yph -> A.xE.yph)

Proof of Theorem hbex

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 |- (E.yph <-> -. A.y -. ph)
2		hbex.1	. . . . 5 |- (ph -> A.xph)
3	2	hbn 1776	. . . 4 |- (-. ph -> A.x -. ph)
4	3	hbal 1736	. . 3 |- (A.y -. ph -> A.xA.y -. ph)
5	4	hbn 1776	. 2 |- (-. A.y -. ph -> A.x -. A.y -. ph)
6	1, 5	hbxfrbi 1568	1 |- (E.yph -> A.xE.yph)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  nfex  1843  19.12OLD  1848  hboprab2  5560  hboprab3  5561  hboprab  5562