Theorem syl2and 354

Description: A syllogism deduction.

Hypotheses

Ref	Expression
syl2and.1	⊢ (φ → ((ψ ∧ χ) → θ))
syl2and.2	⊢ (φ → (τ → ψ))
syl2and.3	⊢ (φ → (η → χ))

Assertion

Ref	Expression
syl2and	⊢ (φ → ((τ ∧ η) → θ))

Proof of Theorem syl2and

Step	Hyp	Ref	Expression
1		syl2and.1	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
2		syl2and.3	. . 3 ⊢ (φ → (η → χ))
3	1, 2	sylan2d 353	. 2 ⊢ (φ → ((ψ ∧ η) → θ))
4		syl2and.2	. 2 ⊢ (φ → (τ → ψ))
5	3, 4	syland 352	1 ⊢ (φ → ((τ ∧ η) → θ))

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  shsvst 5288

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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