Metamath Proof Explorer


Theorem had1

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)

Ref Expression
Assertion had1 φ hadd φ ψ χ ψ χ

Proof

Step Hyp Ref Expression
1 hadrot hadd φ ψ χ hadd ψ χ φ
2 hadbi hadd ψ χ φ ψ χ φ
3 1 2 bitri hadd φ ψ χ ψ χ φ
4 biass hadd φ ψ χ ψ χ φ hadd φ ψ χ ψ χ φ
5 3 4 mpbir hadd φ ψ χ ψ χ φ
6 5 biimpri φ hadd φ ψ χ ψ χ