Theorem nsyl4 159

Description: A negated syllogism inference. (Contributed by NM, 15-Feb-1996.)

Hypotheses

Ref	Expression
nsyl4.1	⊢ (𝜑 → 𝜓)
nsyl4.2	⊢ (¬ 𝜑 → 𝜒)

Assertion

Ref	Expression
nsyl4	⊢ (¬ 𝜒 → 𝜓)

Proof of Theorem nsyl4

Step	Hyp	Ref	Expression
1		nsyl4.2	. . 3 ⊢ (¬ 𝜑 → 𝜒)
2	1	con1i 148	. 2 ⊢ (¬ 𝜒 → 𝜑)
3		nsyl4.1	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	syl 18	1 ⊢ (¬ 𝜒 → 𝜓)

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

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

This theorem is used by:  nsyl5  160  pm2.61i  184  axc7  2347  nfunsn  6912  mptrcl  6991  card2on  9526  carden2a  10018  ax10fromc7  39872  axc5c711  39895  axc5c711to11  39898  naecoms-o  39904  axc5c4c711  45329  axc5c4c711to11  45333  afvco2  48168