Theorem sylancl2 31737

Description: Shortens 5 proofs. (Contributed by BJ, 25-Apr-2019.)

Hypotheses

Ref	Expression
sylancl2.1	⊢ (𝜑 → 𝜓)
sylancl2.2	⊢ 𝜒
sylancl2.3	⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃)

Assertion

Ref	Expression
sylancl2	⊢ (𝜑 → 𝜃)

Proof of Theorem sylancl2

Step	Hyp	Ref	Expression
1		sylancl2.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylancl2.2	. 2 ⊢ 𝜒
3		sylancl2.3	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃)
4	3	biimpi 205	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
5	1, 2, 4	sylancl 693	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383

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 196  df-an 385

This theorem is referenced by: (None)