Theorem triantru3

Description: A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018)

	Ref	Expression
Hypotheses	triantru3.1	$$⊢\phi$$
	triantru3.2	$$⊢\psi$$
Assertion	triantru3	$$⊢\chi \leftrightarrow \left(\phi \wedge \psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		triantru3.1	$$⊢\phi$$
2		triantru3.2	$$⊢\psi$$
3	1	biantrur	$$⊢\left(\psi \wedge \chi \right)\leftrightarrow \left(\phi \wedge \left(\psi \wedge \chi \right)\right)$$
4	2	biantrur	$$⊢\chi \leftrightarrow \left(\psi \wedge \chi \right)$$
5		3anass	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\phi \wedge \left(\psi \wedge \chi \right)\right)$$
6	3 4 5	3bitr4i	$$⊢\chi \leftrightarrow \left(\phi \wedge \psi \wedge \chi \right)$$