Theorem nfsbxy 2002

Description: Similar to hbsb 2009 but with an extra distinct variable constraint, on x and y. (Contributed by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
nfsbxy.1	|- F/ z ph

Assertion

Ref	Expression
nfsbxy	|- F/ z [ y / x ] ph

Distinct variable groups:   x, y   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem nfsbxy

Step	Hyp	Ref	Expression
1		ax-bndl 1562	. 2 |- ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) )
2		nfs1v 1999	. . . 4 |- F/ z [ y / z ] ph
3		drsb1 1852	. . . . 5 |- ( A. z z = x -> ( [ y / z ] ph <-> [ y / x ] ph ) )
4	3	drnf2 1787	. . . 4 |- ( A. z z = x -> ( F/ z [ y / z ] ph <-> F/ z [ y / x ] ph ) )
5	2, 4	mpbii 148	. . 3 |- ( A. z z = x -> F/ z [ y / x ] ph )
6		a16nf 1919	. . . 4 |- ( A. z z = y -> F/ z [ y / x ] ph )
7		df-nf 1514	. . . . . 6 |- ( F/ z x = y <-> A. z ( x = y -> A. z x = y ) )
8	7	albii 1523	. . . . 5 |- ( A. x F/ z x = y <-> A. x A. z ( x = y -> A. z x = y ) )
9		sb5 1942	. . . . . 6 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
10		nfa1 1594	. . . . . . 7 |- F/ x A. x F/ z x = y
11		sp 1564	. . . . . . . 8 |- ( A. x F/ z x = y -> F/ z x = y )
12		nfsbxy.1	. . . . . . . . 9 |- F/ z ph
13	12	a1i 9	. . . . . . . 8 |- ( A. x F/ z x = y -> F/ z ph )
14	11, 13	nfand 1621	. . . . . . 7 |- ( A. x F/ z x = y -> F/ z ( x = y /\ ph ) )
15	10, 14	nfexd 1814	. . . . . 6 |- ( A. x F/ z x = y -> F/ z E. x ( x = y /\ ph ) )
16	9, 15	nfxfrd 1528	. . . . 5 |- ( A. x F/ z x = y -> F/ z [ y / x ] ph )
17	8, 16	sylbir 135	. . . 4 |- ( A. x A. z ( x = y -> A. z x = y ) -> F/ z [ y / x ] ph )
18	6, 17	jaoi 728	. . 3 |- ( ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) -> F/ z [ y / x ] ph )
19	5, 18	jaoi 728	. 2 |- ( ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) ) -> F/ z [ y / x ] ph )
20	1, 19	ax-mp 5	1 |- F/ z [ y / x ] ph

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   A.wal 1400   F/wnf 1513   E.wex 1545   [wsb 1815

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  nfsb  2006  sbalyz  2059  opelopabsb  4397  bezoutlemmain  12753