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  13622  psdmul  22104  efrlim  26931  addsval  27921  mulscan2d  28134  dvdsruasso  33400  ssdifidlprm  33473  fsuppssind  42616  tfsconcat0i  43369  oadif1lem  43403  oadif1  43404  reabsifneg  43656  natglobalincr  46906  f1cof1b  47106  isubgr3stgrlem6  47983  prsthinc  49350