Theorem nfsbxy 1859

Description: Similar to hbsb 1864 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 1439	. 2 |- ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) )
2		nfs1v 1856	. . . 4 |- F/ z [ y / z ] ph
3		drsb1 1720	. . . . 5 |- ( A. z z = x -> ( [ y / z ] ph <-> [ y / x ] ph ) )
4	3	drnf2 1662	. . . 4 |- ( A. z z = x -> ( F/ z [ y / z ] ph <-> F/ z [ y / x ] ph ) )
5	2, 4	mpbii 146	. . 3 |- ( A. z z = x -> F/ z [ y / x ] ph )
6		a16nf 1787	. . . 4 |- ( A. z z = y -> F/ z [ y / x ] ph )
7		df-nf 1390	. . . . . 6 |- ( F/ z x = y <-> A. z ( x = y -> A. z x = y ) )
8	7	albii 1399	. . . . 5 |- ( A. x F/ z x = y <-> A. x A. z ( x = y -> A. z x = y ) )
9		sb5 1808	. . . . . 6 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
10		nfa1 1474	. . . . . . 7 |- F/ x A. x F/ z x = y
11		sp 1441	. . . . . . . 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 1500	. . . . . . 7 |- ( A. x F/ z x = y -> F/ z ( x = y /\ ph ) )
15	10, 14	nfexd 1684	. . . . . 6 |- ( A. x F/ z x = y -> F/ z E. x ( x = y /\ ph ) )
16	9, 15	nfxfrd 1404	. . . . 5 |- ( A. x F/ z x = y -> F/ z [ y / x ] ph )
17	8, 16	sylbir 133	. . . 4 |- ( A. x A. z ( x = y -> A. z x = y ) -> F/ z [ y / x ] ph )
18	6, 17	jaoi 668	. . 3 |- ( ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) -> F/ z [ y / x ] ph )
19	5, 18	jaoi 668	. 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 7	1 |- F/ z [ y / x ] ph

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   A.wal 1282   F/wnf 1389   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  nfsb  1863  sbalyz  1916  opelopabsb  4015  bezoutlemmain  10387