Theorem sbal 2462

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 2316	. . . 4 |- F/ y A. x x = z
2		axc16gb 2136	. . . 4 |- ( A. x x = z -> ( ph <-> A. x ph ) )
3	1, 2	sbbid 2403	. . 3 |- ( A. x x = z -> ( [ z / y ] ph <-> [ z / y ] A. x ph ) )
4		axc16gb 2136	. . 3 |- ( A. x x = z -> ( [ z / y ] ph <-> A. x [ z / y ] ph ) )
5	3, 4	bitr3d 270	. 2 |- ( A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
6		sbal1 2460	. 2 |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
7	5, 6	pm2.61i 176	1 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )

Syntax hints:   <-> wb 196   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbex  2463  sbalv  2464  sbcal  3485  ax11-pm2  32823  bj-sbnf  32828  sbcalgOLD  38752