Theorem syl9r 67

Description: A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
syl9r.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl9r.2	⊢ (𝜃 → (𝜒 → 𝜏))

Assertion

Ref	Expression
syl9r	⊢ (𝜃 → (𝜑 → (𝜓 → 𝜏)))

Proof of Theorem syl9r

Step	Hyp	Ref	Expression
1		syl9r.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		syl9r.2	. . 3 ⊢ (𝜃 → (𝜒 → 𝜏))
3	1, 2	syl9 66	. 2 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏)))
4	3	com12 27	1 ⊢ (𝜃 → (𝜑 → (𝜓 → 𝜏)))

Syntax hints:   → wi 4

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

This theorem is referenced by:  sylan9r  390  pm2.85dc  811  looinvdc  821  pclem6  1265  nfimd  1477  19.23t  1567  fununi  4967  dfimafn  5222  funimass3  5283  nnsub  7952