Theorem cnf1dd 38097

Description: A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)

Hypotheses

Ref	Expression
cnf1dd.1	⊢ (𝜑 → (𝜓 → ¬ 𝜒))
cnf1dd.2	⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))

Assertion

Ref	Expression
cnf1dd	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem cnf1dd

Step	Hyp	Ref	Expression
1		cnf1dd.1	. . 3 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
2		cnf1dd.2	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))
3	1, 2	jcad 512	. 2 ⊢ (𝜑 → (𝜓 → (¬ 𝜒 ∧ (𝜒 ∨ 𝜃))))
4		df-or 849	. . 3 ⊢ ((𝜒 ∨ 𝜃) ↔ (¬ 𝜒 → 𝜃))
5		pm3.35 803	. . 3 ⊢ ((¬ 𝜒 ∧ (¬ 𝜒 → 𝜃)) → 𝜃)
6	4, 5	sylan2b 594	. 2 ⊢ ((¬ 𝜒 ∧ (𝜒 ∨ 𝜃)) → 𝜃)
7	3, 6	syl6 35	1 ⊢ (𝜑 → (𝜓 → 𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848

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  df-or 849

This theorem is referenced by:  cnf2dd  38098  cnfn1dd  38099  mpobi123f  38169  mptbi12f  38173  ac6s6  38179