Theorem mpbiran2d 709

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 708	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  5950  funsnfsupp  9298  discld  23064  cncfcdm  24875  itgfsum  25804  dchreq  27235  lgsneg  27298  lgsquadlem2  27358  z12bdaylem1  28476  dfconngr1  30273  cover2  38050  iscnrm3rlem6  49432  0funcglem  49570  0funcg2  49571  thincmon  49920  thincepi  49921