Theorem mpnanrd 415

Description: Eliminate the right side of a negated conjunction in an implication. (Contributed by ML, 17-Oct-2020.)

Hypotheses

Ref	Expression
mpnanrd.1	⊢ (𝜑 → 𝜓)
mpnanrd.2	⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
mpnanrd	⊢ (𝜑 → ¬ 𝜒)

Proof of Theorem mpnanrd

Step	Hyp	Ref	Expression
1		mpnanrd.1	. 2 ⊢ (𝜑 → 𝜓)
2		mpnanrd.2	. . 3 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
3		imnan 405	. . 3 ⊢ ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒))
4	2, 3	sylibr 237	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
5	1, 4	mpd 16	1 ⊢ (𝜑 → ¬ 𝜒)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  ecase2d  1047  mxidlirred  33997  fedgmullem2  34262  onsucuni3  38290  hashnexinj  43178  veroquaddetzerod  50985