Theorem sylibda 39673

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

Hypotheses

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

Assertion

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

Proof of Theorem sylibda

Step	Hyp	Ref	Expression
1		sylibda.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	biimpa 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		sylibda.2	. 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
4	2, 3	syldan 595	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:  fsuppssind  39772