Theorem sbalv 1881

Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
sbalv.1	|- ( [ y / x ] ph <-> ps )

Assertion

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

Distinct variable groups:   x, z   y, z

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

Proof of Theorem sbalv

Step	Hyp	Ref	Expression
1		sbal 1876	. 2 |- ( [ y / x ] A. z ph <-> A. z [ y / x ] ph )
2		sbalv.1	. . 3 |- ( [ y / x ] ph <-> ps )
3	2	albii 1359	. 2 |- ( A. z [ y / x ] ph <-> A. z ps )
4	1, 3	bitri 173	1 |- ( [ y / x ] A. z ph <-> A. z ps )

Syntax hints:   <-> wb 98   A.wal 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbmo  1959  sbabel  2203  peano2  4318