Theorem 19.26 1492

Description: Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Assertion

Ref	Expression
19.26	|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )

Proof of Theorem 19.26

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( ph /\ ps ) -> ph )
2	1	alimi 1466	. . 3 |- ( A. x ( ph /\ ps ) -> A. x ph )
3		simpr 110	. . . 4 |- ( ( ph /\ ps ) -> ps )
4	3	alimi 1466	. . 3 |- ( A. x ( ph /\ ps ) -> A. x ps )
5	2, 4	jca 306	. 2 |- ( A. x ( ph /\ ps ) -> ( A. x ph /\ A. x ps ) )
6		id 19	. . 3 |- ( ( ph /\ ps ) -> ( ph /\ ps ) )
7	6	alanimi 1470	. 2 |- ( ( A. x ph /\ A. x ps ) -> A. x ( ph /\ ps ) )
8	5, 7	impbii 126	1 |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.26-2  1493  19.26-3an  1494  albiim  1498  2albiim  1499  hband  1500  hban  1558  19.27h  1571  19.27  1572  19.28h  1573  19.28  1574  nford  1578  nfand  1579  equsexd  1740  equveli  1770  sbanv  1901  2eu4  2135  bm1.1  2178  r19.26m  2625  unss  3334  ralunb  3341  ssin  3382  intun  3902  intpr  3903  eqrelrel  4761  relop  4813  eqoprab2b  5977  dfer2  6590  omniwomnimkv  7228