Theorem bian1d 32475

Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.) (Proof shortened by Hongxiu Chen, 29-Jun-2025.)

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		ibar 528	. . . . 5 ⊢ (𝜒 → (𝜃 ↔ (𝜒 ∧ 𝜃)))
3	2	bicomd 223	. . . 4 ⊢ (𝜒 → ((𝜒 ∧ 𝜃) ↔ 𝜃))
4	1, 3	sylan9bb 509	. . 3 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))
5	4	ex 412	. 2 ⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜃)))
6	5	pm5.32d 577	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  funcnvmpt  32677