Theorem had1 1542

Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Assertion

Ref	Expression
had1	|- ( ph -> (hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) )

Proof of Theorem had1

Step	Hyp	Ref	Expression
1		hadrot 1540	. . . 4 |- (hadd ( ph , ps , ch ) <-> hadd ( ps , ch , ph ) )
2		hadbi 1537	. . . 4 |- (hadd ( ps , ch , ph ) <-> ( ( ps <-> ch ) <-> ph ) )
3	1, 2	bitri 264	. . 3 |- (hadd ( ph , ps , ch ) <-> ( ( ps <-> ch ) <-> ph ) )
4		biass 374	. . 3 |- ( ( (hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) <-> ph ) <-> (hadd ( ph , ps , ch ) <-> ( ( ps <-> ch ) <-> ph ) ) )
5	3, 4	mpbir 221	. 2 |- ( (hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) <-> ph )
6	5	biimpri 218	1 |- ( ph -> (hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196  haddwhad 1532

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 197  df-xor 1465  df-had 1533

This theorem is referenced by:  had0  1543  hadifp  1544  sadadd2lem2  15172