Theorem sbalyz 2052

Description: Move universal quantifier in and out of substitution. Identical to sbal 2053 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 1589	. . . 4 |- F/ x A. x ph
2	1	nfsbxy 1995	. . 3 |- F/ x [ z / y ] A. x ph
3		ax-4 1558	. . . 4 |- ( A. x ph -> ph )
4	3	sbimi 1812	. . 3 |- ( [ z / y ] A. x ph -> [ z / y ] ph )
5	2, 4	alrimi 1570	. 2 |- ( [ z / y ] A. x ph -> A. x [ z / y ] ph )
6		sb6 1935	. . . . 5 |- ( [ z / y ] ph <-> A. y ( y = z -> ph ) )
7	6	albii 1518	. . . 4 |- ( A. x [ z / y ] ph <-> A. x A. y ( y = z -> ph ) )
8		alcom 1526	. . . 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 1576	. . . . . 6 |- F/ x y = z
11	10	stdpc5 1632	. . . . 5 |- ( A. x ( y = z -> ph ) -> ( y = z -> A. x ph ) )
12	11	alimi 1503	. . . 4 |- ( A. y A. x ( y = z -> ph ) -> A. y ( y = z -> A. x ph ) )
13		sb2 1815	. . . 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 1395   [wsb 1810

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sbal  2053