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  13505  chnccat  18532  psdmul  22081  efrlim  26906  addsval  27905  mulscan2d  28118  dvdsruasso  33350  ssdifidlprm  33423  fsuppssind  42696  tfsconcat0i  43448  oadif1lem  43482  oadif1  43483  reabsifneg  43735  natglobalincr  46985  f1cof1b  47187  isubgr3stgrlem6  48081  prsthinc  49575