Theorem sbalyz 1917

Description: Move universal quantifier in and out of substitution. Identical to sbal 1918 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 1475	. . . 4 |- F/ x A. x ph
2	1	nfsbxy 1860	. . 3 |- F/ x [ z / y ] A. x ph
3		ax-4 1441	. . . 4 |- ( A. x ph -> ph )
4	3	sbimi 1688	. . 3 |- ( [ z / y ] A. x ph -> [ z / y ] ph )
5	2, 4	alrimi 1456	. 2 |- ( [ z / y ] A. x ph -> A. x [ z / y ] ph )
6		sb6 1808	. . . . 5 |- ( [ z / y ] ph <-> A. y ( y = z -> ph ) )
7	6	albii 1400	. . . 4 |- ( A. x [ z / y ] ph <-> A. x A. y ( y = z -> ph ) )
8		alcom 1408	. . . 4 |- ( A. x A. y ( y = z -> ph ) <-> A. y A. x ( y = z -> ph ) )
9	7, 8	bitri 182	. . 3 |- ( A. x [ z / y ] ph <-> A. y A. x ( y = z -> ph ) )
10		nfv 1462	. . . . . 6 |- F/ x y = z
11	10	stdpc5 1517	. . . . 5 |- ( A. x ( y = z -> ph ) -> ( y = z -> A. x ph ) )
12	11	alimi 1385	. . . 4 |- ( A. y A. x ( y = z -> ph ) -> A. y ( y = z -> A. x ph ) )
13		sb2 1691	. . . 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 119	. 2 |- ( A. x [ z / y ] ph -> [ z / y ] A. x ph )
16	5, 15	impbii 124	1 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   [wsb 1686

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687

This theorem is referenced by:  sbal  1918