Theorem mpbiran2d 720

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 468	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒)))
4	1, 3	mpbirand 719	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400

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 210  df-an 401

This theorem is referenced by:  opelidres  5988  funsnfsupp  9348  discld  23211  cncfcdm  25022  itgfsum  25951  dchreq  27384  lgsneg  27447  lgsquadlem2  27507  z12bdaylem1  28625  lnincplng  29020  dfconngr1  30476  cover2  38249  iscnrm3rlem6  49601  0funcglem  49739  0funcg2  49740  thincmon  50089  thincepi  50090