Theorem syl2ani 358

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl2ani.1	⊢ (φ → ((ψ ∧ χ) → θ))
syl2ani.2	⊢ (τ → ψ)
syl2ani.3	⊢ (η → χ)

Assertion

Ref	Expression
syl2ani	⊢ (φ → ((τ ∧ η) → θ))

Proof of Theorem syl2ani

Step	Hyp	Ref	Expression
1		syl2ani.1	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
2		syl2ani.3	. . 3 ⊢ (η → χ)
3	1, 2	sylan2i 357	. 2 ⊢ (φ → ((ψ ∧ η) → θ))
4		syl2ani.2	. 2 ⊢ (τ → ψ)
5	3, 4	sylani 356	1 ⊢ (φ → ((τ ∧ η) → θ))

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  sqrle 4765

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

This theorem depends on definitions:  df-bi 128  df-an 198

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