Theorem sylbida 603

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 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		sylbida.2	. 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
4	2, 3	syldan 602	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

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:  fzdif1  13633  chnccat  18682  ssdifidlprm  21455  psdmul  22298  efrlim  27100  addsval  28121  mulscan2d  28338  dvdsruasso  33642  fsuppssind  43251  tfsconcat0i  43998  oadif1lem  44032  oadif1  44033  reabsifneg  44284  natglobalincr  47519  f1cof1b  47737  nprmdvdsfacm1lem4  48298  isubgr3stgrlem6  48659  prsthinc  50161