Theorem sbal 1924

Description: Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)

Assertion

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

Distinct variable groups:   x, y   x, z

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

Proof of Theorem sbal

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbalyz 1923	. . . 4 |- ( [ w / y ] A. x ph <-> A. x [ w / y ] ph )
2	1	sbbii 1695	. . 3 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / w ] A. x [ w / y ] ph )
3		sbalyz 1923	. . 3 |- ( [ z / w ] A. x [ w / y ] ph <-> A. x [ z / w ] [ w / y ] ph )
4	2, 3	bitri 182	. 2 |- ( [ z / w ] [ w / y ] A. x ph <-> A. x [ z / w ] [ w / y ] ph )
5		ax-17 1464	. . 3 |- ( A. x ph -> A. w A. x ph )
6	5	sbco2v 1869	. 2 |- ( [ z / w ] [ w / y ] A. x ph <-> [ z / y ] A. x ph )
7		ax-17 1464	. . . 4 |- ( ph -> A. w ph )
8	7	sbco2v 1869	. . 3 |- ( [ z / w ] [ w / y ] ph <-> [ z / y ] ph )
9	8	albii 1404	. 2 |- ( A. x [ z / w ] [ w / y ] ph <-> A. x [ z / y ] ph )
10	4, 6, 9	3bitr3i 208	1 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Syntax hints:   <-> wb 103   A.wal 1287   [wsb 1692

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693

This theorem is referenced by:  sbal1  1926  sbalv  1929  sbcal  2890  sbcalg  2891