Theorem syl9 72

Description: A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)

Hypotheses

Ref	Expression
syl9.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl9.2	⊢ (𝜃 → (𝜒 → 𝜏))

Assertion

Ref	Expression
syl9	⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏)))

Proof of Theorem syl9

Step	Hyp	Ref	Expression
1		syl9.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		syl9.2	. . 3 ⊢ (𝜃 → (𝜒 → 𝜏))
3	2	a1i 9	. 2 ⊢ (𝜑 → (𝜃 → (𝜒 → 𝜏)))
4	1, 3	syl5d 68	1 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏)))

Syntax hints:   → wi 4

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

This theorem is referenced by:  syl9r  73  com23  78  sylan9  409  pm4.79dc  904  pclem6  1385  bilukdc  1407  sbequi  1853  reuss2  3443  reupick  3447  elres  4982  funimass4  5611  fliftfun  5843  elabgf2  15426  bj-rspgt  15432