Theorem hadbi 1500

Description: The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
hadbi	|- (hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> ch ) )

Proof of Theorem hadbi

Step	Hyp	Ref	Expression
1		df-xor 1405	. 2 |- ( ( ( ph \/_ ps ) \/_ ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
2		df-had 1496	. 2 |- (hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) )
3		xnor 1406	. . . 4 |- ( ( ph <-> ps ) <-> -. ( ph \/_ ps ) )
4	3	bibi1i 316	. . 3 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( -. ( ph \/_ ps ) <-> ch ) )
5		nbbn 360	. . 3 |- ( ( -. ( ph \/_ ps ) <-> ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
6	4, 5	bitri 253	. 2 |- ( ( ( ph <-> ps ) <-> ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
7	1, 2, 6	3bitr4i 281	1 |- (hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> ch ) )

Syntax hints:   -. wn 3   <-> wb 188   \/_ wxo 1404  haddwhad 1495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 189  df-xor 1405  df-had 1496

This theorem is referenced by:  hadnot  1504  had1  1505