Theorem sbal 2206

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

Distinct variable groups:   ,   ,

Proof of Theorem sbal

Step	Hyp	Ref	Expression
1		nfae 2056	. . . 4
2		ax16gb 1942	. . . 4
3	1, 2	sbbid 2144	. . 3
4		ax16gb 1942	. . 3
5	3, 4	bitr3d 255	. 2
6		sbal1 2204	. 2
7	5, 6	pm2.61i 164	1

Syntax hints:  <->wb 184  A.wal 1393  [wsb 1739

This theorem is referenced by:  sbex  2207  sbalv  2208  sbcal  3379  sbcalgOLD  3380  ax11-pm2  34409  bj-sbnf  34412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740