Theorem sylan9 638

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypotheses

Ref	Expression
sylan9.1	⊢ (φ → (ψ → χ))
sylan9.2	⊢ (θ → (χ → τ))

Assertion

Ref	Expression
sylan9	⊢ ((φ ∧ θ) → (ψ → τ))

Proof of Theorem sylan9

Step	Hyp	Ref	Expression
1		sylan9.1	. . 3 ⊢ (φ → (ψ → χ))
2		sylan9.2	. . 3 ⊢ (θ → (χ → τ))
3	1, 2	syl9 66	. 2 ⊢ (φ → (θ → (ψ → τ)))
4	3	imp 418	1 ⊢ ((φ ∧ θ) → (ψ → τ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358

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

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  sbequi  2059  sbal1  2126  rspc2  2961  rspc3v  2965  copsexg  4608  chfnrn  5400  ffnfv  5428  f1elima  5475  isotr  5496