Theorem nfexd 1854

Description: If x is not free in ph, it is not free in E.yph. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
nfald.1	|- F/yph
nfald.2	|- (ph -> F/xps)

Assertion

Ref	Expression
nfexd	|- (ph -> F/xE.yps)

Proof of Theorem nfexd

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 |- (E.yps <-> -. A.y -. ps)
2		nfald.1	. . . 4 |- F/yph
3		nfald.2	. . . . 5 |- (ph -> F/xps)
4	3	nfnd 1791	. . . 4 |- (ph -> F/x -. ps)
5	2, 4	nfald 1852	. . 3 |- (ph -> F/xA.y -. ps)
6	5	nfnd 1791	. 2 |- (ph -> F/x -. A.y -. ps)
7	1, 6	nfxfrd 1571	1 |- (ph -> F/xE.yps)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541   F/wnf 1544

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  df-nf 1545

This theorem is referenced by:  nfeld  2505