Theorem sylbida 604

Description: A syllogism deduction. (Contributed by SN, 16-Jul-2024.)

Hypotheses

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

Assertion

Ref	Expression
sylbida	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Proof of Theorem sylbida

Step	Hyp	Ref	Expression
1		sylbida.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	biimpa 482	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		sylbida.2	. 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
4	2, 3	syldan 603	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  fzdif1  13739  chnccat  18800  ssdifidlprm  21642  psdmul  22487  efrlim  27297  addsval  28348  mulscan2d  28565  dvdsruasso  33940  fsuppssind  43621  prjspnnorm  43661  tfsconcat0i  44346  oadif1lem  44380  oadif1  44381  reabsifneg  44631  f1cof1b  48146  nprmdvdsfacm1lem4  48707  isubgr3stgrlem6  49068  prsthinc  50571