Theorem mpbirand 530

Description: Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

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

Assertion

Ref	Expression
mpbirand	⊢ (𝜑 → (𝜓 ↔ 𝜃))

Proof of Theorem mpbirand

Step	Hyp	Ref	Expression
1		mpbirand.2	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
2		mpbirand.1	. . 3 ⊢ (𝜑 → 𝜒)
3	2	biantrurd 529	. 2 ⊢ (𝜑 → (𝜃 ↔ (𝜒 ∧ 𝜃)))
4	1, 3	bitr4d 271	1 ⊢ (𝜑 → (𝜓 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  3anibar  1229  rmob2  3531  opbrop  5198  opelresi  5408  iscvs  22927  isspthonpth  26645  esum2dlem  30154  ntrclselnel1  38355  ntrneicls00  38387  vonvolmbl  40875