Theorem sbalv 2129

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.zph <-> A.zps)

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 2127	. 2 |- ([y / x]A.zph <-> A.z[y / x]ph)
2		sbalv.1	. . 3 |- ([y / x]ph <-> ps)
3	2	albii 1566	. 2 |- (A.z[y / x]ph <-> A.zps)
4	1, 3	bitri 240	1 |- ([y / x]A.zph <-> A.zps)

Syntax hints:   <-> wb 176  A.wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbmo  2234  sbabel  2516