Theorem bian1d 30216

Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)

Hypothesis

Ref	Expression
bian1d.1	⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))

Assertion

Ref	Expression
bian1d	⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Proof of Theorem bian1d

Step	Hyp	Ref	Expression
1		bian1d.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
2	1	biimpd 231	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃)))
3	2	adantld 493	. 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) → (𝜒 ∧ 𝜃)))
4		simpl 485	. . . 4 ⊢ ((𝜒 ∧ 𝜃) → 𝜒)
5	4	a1i 11	. . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜒))
6	1	biimprd 250	. . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜓))
7	5, 6	jcad 515	. 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → (𝜒 ∧ 𝜓)))
8	3, 7	impbid 214	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  funcnvmpt  30404