Theorem sbal 2196

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 2029	. . . 4 |- F/ y A. x x = z
2		ax16gb 1889	. . . 4 |- ( A. x x = z -> ( ph <-> A. x ph ) )
3	1, 2	sbbid 2118	. . 3 |- ( A. x x = z -> ( [ z / y ] ph <-> [ z / y ] A. x ph ) )
4		ax16gb 1889	. . 3 |- ( A. x x = z -> ( [ z / y ] ph <-> A. x [ z / y ] ph ) )
5	3, 4	bitr3d 255	. 2 |- ( A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
6		sbal1 2193	. 2 |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
7	5, 6	pm2.61i 164	1 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Syntax hints:   <-> wb 184   A.wal 1377   [wsb 1711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712

This theorem is referenced by:  sbex  2198  sbalv  2199  sbcal  3388  sbcalgOLD  3389  ax11-pm2  33891  bj-sbnf  33894