Theorem sylbida 592

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 476	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		sylbida.2	. 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
4	2, 3	syldan 591	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

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:  fzdif1  13521  chnccat  18549  psdmul  22109  efrlim  26935  addsval  27958  mulscan2d  28175  dvdsruasso  33466  ssdifidlprm  33539  fsuppssind  42846  tfsconcat0i  43597  oadif1lem  43631  oadif1  43632  reabsifneg  43883  natglobalincr  47131  f1cof1b  47333  isubgr3stgrlem6  48227  prsthinc  49719