Theorem mpbiran2d 707

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.1	. 2 ⊢ (𝜑 → 𝜃)
2		mpbiran2d.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
3	2	biancomd 463	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒)))
4	1, 3	mpbirand 706	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

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:  opelidres  6021  funsnfsupp  9461  discld  23118  cncfcdm  24943  itgfsum  25882  dchreq  27320  lgsneg  27383  lgsquadlem2  27443  dfconngr1  30220  cover2  37675  iscnrm3rlem6  48625  thincmon  48701  thincepi  48702