Theorem sbalv 2021

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 2016	. 2 |- ( [ y / x ] A. z ph <-> A. z [ y / x ] ph )
2		sbalv.1	. . 3 |- ( [ y / x ] ph <-> ps )
3	2	albii 1481	. 2 |- ( A. z [ y / x ] ph <-> A. z ps )
4	1, 3	bitri 184	1 |- ( [ y / x ] A. z ph <-> A. z ps )

Syntax hints:   <-> wb 105   A.wal 1362   [wsb 1773

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbmo  2101  sbabel  2363  peano2  4627