Theorem triantru3 39169

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

Hypotheses

Ref	Expression
triantru3.1	⊢ 𝜑
triantru3.2	⊢ 𝜓

Assertion

Ref	Expression
triantru3	⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))

Proof of Theorem triantru3

Step	Hyp	Ref	Expression
1		triantru3.1	. . 3 ⊢ 𝜑
2	1	biantrur 540	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
3		triantru3.2	. . 3 ⊢ 𝜓
4	3	biantrur 540	. 2 ⊢ (𝜒 ↔ (𝜓 ∧ 𝜒))
5		3anass 1111	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
6	2, 4, 5	3bitr4i 306	1 ⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  eqvrelcoss  39634  eqvrelcoss3  39635