Theorem 4exdistr 1781

Description: Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)

Assertion

Ref	Expression
4exdistr

Distinct variable groups:   ,   ,   ,   ,   ,   ,

Proof of Theorem 4exdistr

Step	Hyp	Ref	Expression
1		19.42v 1775	. . . . 5
2	1	anbi2i 694	. . . 4
3		19.42v 1775	. . . 4
4		df-3an 975	. . . 4
5	2, 3, 4	3bitr4i 277	. . 3
6	5	3exbii 1669	. 2
7		3exdistr 1780	. 2
8	6, 7	bitri 249	1

Syntax hints:  <->wb 184  /\wa 369  /\w3a 973  E.wex 1612

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

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1613