Theorem syld3an1 873

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	⊢ ((φ ⋀ ψ ⋀ χ) → θ)
syld3an1.2	⊢ ((τ ⋀ ψ ⋀ χ) → φ)

Assertion

Ref	Expression
syld3an1	⊢ ((τ ⋀ ψ ⋀ χ) → θ)

Proof of Theorem syld3an1

Step	Hyp	Ref	Expression
1		syl3an.1	. . . 4 ⊢ ((φ ⋀ ψ ⋀ χ) → θ)
2	1	3com13 840	. . 3 ⊢ ((χ ⋀ ψ ⋀ φ) → θ)
3		syld3an1.2	. . . 4 ⊢ ((τ ⋀ ψ ⋀ χ) → φ)
4	3	3com13 840	. . 3 ⊢ ((χ ⋀ ψ ⋀ τ) → φ)
5	2, 4	syld3an3 872	. 2 ⊢ ((χ ⋀ ψ ⋀ τ) → θ)
6	5	3com13 840	1 ⊢ ((τ ⋀ ψ ⋀ χ) → θ)

Syntax hints:   → wi 3   ⋀ w3a 777

This theorem is referenced by:  npncant 5419  ppncant 5500

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779

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