Theorem sbalyz 2050

Description: Move universal quantifier in and out of substitution. Identical to sbal 2051 except that it has an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 29-Dec-2017.)

Assertion

Ref	Expression
sbalyz	|- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Distinct variable group:   x, y, z

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

Proof of Theorem sbalyz

Step	Hyp	Ref	Expression
1		nfa1 1587	. . . 4 |- F/ x A. x ph
2	1	nfsbxy 1993	. . 3 |- F/ x [ z / y ] A. x ph
3		ax-4 1556	. . . 4 |- ( A. x ph -> ph )
4	3	sbimi 1810	. . 3 |- ( [ z / y ] A. x ph -> [ z / y ] ph )
5	2, 4	alrimi 1568	. 2 |- ( [ z / y ] A. x ph -> A. x [ z / y ] ph )
6		sb6 1933	. . . . 5 |- ( [ z / y ] ph <-> A. y ( y = z -> ph ) )
7	6	albii 1516	. . . 4 |- ( A. x [ z / y ] ph <-> A. x A. y ( y = z -> ph ) )
8		alcom 1524	. . . 4 |- ( A. x A. y ( y = z -> ph ) <-> A. y A. x ( y = z -> ph ) )
9	7, 8	bitri 184	. . 3 |- ( A. x [ z / y ] ph <-> A. y A. x ( y = z -> ph ) )
10		nfv 1574	. . . . . 6 |- F/ x y = z
11	10	stdpc5 1630	. . . . 5 |- ( A. x ( y = z -> ph ) -> ( y = z -> A. x ph ) )
12	11	alimi 1501	. . . 4 |- ( A. y A. x ( y = z -> ph ) -> A. y ( y = z -> A. x ph ) )
13		sb2 1813	. . . 4 |- ( A. y ( y = z -> A. x ph ) -> [ z / y ] A. x ph )
14	12, 13	syl 14	. . 3 |- ( A. y A. x ( y = z -> ph ) -> [ z / y ] A. x ph )
15	9, 14	sylbi 121	. 2 |- ( A. x [ z / y ] ph -> [ z / y ] A. x ph )
16	5, 15	impbii 126	1 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   [wsb 1808

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sbal  2051