Theorem nsyl4 155

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 143	. 2 ⊢ (¬ 𝜒 → 𝜑)
3		nsyl4.1	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	syl 17	1 ⊢ (¬ 𝜒 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  pm5.55  937  axc7  2117  moanim  2517  moexex  2529  nfunsn  6135  mptrcl  6198  card2on  8342  carden2a  8675  wwlknfi  26266  ax10fromc7  33198  axc5c711  33221  axc5c711to11  33224  naecoms-o  33230  axc5c4c711  37624  axc5c4c711to11  37628  afvco2  39905  wwlksnfi  41112