Theorem sbal 2311

Description: Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-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

Step	Hyp	Ref	Expression
1		nfae 2165	. . . 4 |- F/ y A. x x = z
2		axc16gb 2044	. . . 4 |- ( A. x x = z -> ( ph <-> A. x ph ) )
3	1, 2	sbbid 2252	. . 3 |- ( A. x x = z -> ( [ z / y ] ph <-> [ z / y ] A. x ph ) )
4		axc16gb 2044	. . 3 |- ( A. x x = z -> ( [ z / y ] ph <-> A. x [ z / y ] ph ) )
5	3, 4	bitr3d 263	. 2 |- ( A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
6		sbal1 2309	. 2 |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
7	5, 6	pm2.61i 169	1 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Syntax hints:   <-> wb 189   A.wal 1450   [wsb 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806

This theorem is referenced by:  sbex  2312  sbalv  2313  sbcal  3305  ax11-pm2  31504  bj-sbnf  31509  sbcalgOLD  36973