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Theorem cnf1dd 32856
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 → (𝜒𝜃)))
Assertion
Ref Expression
cnf1dd (𝜑 → (𝜓𝜃))

Proof of Theorem cnf1dd
StepHypRef Expression
1 cnf1dd.1 . . 3 (𝜑 → (𝜓 → ¬ 𝜒))
2 cnf1dd.2 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
31, 2jcad 554 . 2 (𝜑 → (𝜓 → (¬ 𝜒 ∧ (𝜒𝜃))))
4 df-or 384 . . 3 ((𝜒𝜃) ↔ (¬ 𝜒𝜃))
5 pm3.35 609 . . 3 ((¬ 𝜒 ∧ (¬ 𝜒𝜃)) → 𝜃)
64, 5sylan2b 491 . 2 ((¬ 𝜒 ∧ (𝜒𝜃)) → 𝜃)
73, 6syl6 34 1 (𝜑 → (𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  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-or 384  df-an 385
This theorem is referenced by:  cnf2dd  32857  cnfn1dd  32858  mpt2bi123f  32935  mptbi12f  32939  ac6s6  32944
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