Theorem sbalyz 1975

Description: Move universal quantifier in and out of substitution. Identical to sbal 1976 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 1522	. . . 4 |- F/ x A. x ph
2	1	nfsbxy 1916	. . 3 |- F/ x [ z / y ] A. x ph
3		ax-4 1488	. . . 4 |- ( A. x ph -> ph )
4	3	sbimi 1738	. . 3 |- ( [ z / y ] A. x ph -> [ z / y ] ph )
5	2, 4	alrimi 1503	. 2 |- ( [ z / y ] A. x ph -> A. x [ z / y ] ph )
6		sb6 1859	. . . . 5 |- ( [ z / y ] ph <-> A. y ( y = z -> ph ) )
7	6	albii 1447	. . . 4 |- ( A. x [ z / y ] ph <-> A. x A. y ( y = z -> ph ) )
8		alcom 1455	. . . 4 |- ( A. x A. y ( y = z -> ph ) <-> A. y A. x ( y = z -> ph ) )
9	7, 8	bitri 183	. . 3 |- ( A. x [ z / y ] ph <-> A. y A. x ( y = z -> ph ) )
10		nfv 1509	. . . . . 6 |- F/ x y = z
11	10	stdpc5 1564	. . . . 5 |- ( A. x ( y = z -> ph ) -> ( y = z -> A. x ph ) )
12	11	alimi 1432	. . . 4 |- ( A. y A. x ( y = z -> ph ) -> A. y ( y = z -> A. x ph ) )
13		sb2 1741	. . . 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 120	. 2 |- ( A. x [ z / y ] ph -> [ z / y ] A. x ph )
16	5, 15	impbii 125	1 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  sbal  1976