Theorem mpbiran2d 442

Description: Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.)

Hypotheses

Ref	Expression
mpbiran2d.1	⊢ (𝜑 → 𝜃)
mpbiran2d.2	⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))

Assertion

Ref	Expression
mpbiran2d	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem mpbiran2d

Step	Hyp	Ref	Expression
1		mpbiran2d.2	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
2		mpbiran2d.1	. . 3 ⊢ (𝜑 → 𝜃)
3	2	biantrud 304	. 2 ⊢ (𝜑 → (𝜒 ↔ (𝜒 ∧ 𝜃)))
4	1, 3	bitr4d 191	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  lgsneg  14565