Theorem sbalv 1993

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 1988	. 2 |- ( [ y / x ] A. z ph <-> A. z [ y / x ] ph )
2		sbalv.1	. . 3 |- ( [ y / x ] ph <-> ps )
3	2	albii 1458	. 2 |- ( A. z [ y / x ] ph <-> A. z ps )
4	1, 3	bitri 183	1 |- ( [ y / x ] A. z ph <-> A. z ps )

Syntax hints:   <-> wb 104   A.wal 1341   [wsb 1750

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbmo  2073  sbabel  2335  peano2  4572