Theorem sylani 356

Description: A syllogism inference.

Hypotheses

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

Assertion

Ref	Expression
sylani	⊢ (φ → ((τ ∧ χ) → θ))

Proof of Theorem sylani

Step	Hyp	Ref	Expression
1		sylani.1	. 2 ⊢ (φ → ((ψ ∧ χ) → θ))
2		sylani.2	. . 3 ⊢ (τ → ψ)
3	2	a1i 7	. 2 ⊢ (φ → (τ → ψ))
4	1, 3	syland 352	1 ⊢ (φ → ((τ ∧ χ) → θ))

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  syl2ani 358  inf3lem2 3465  zornlem5 3607  distrlem4pr 3924  uzwo 4605  projlem1 5193  projlem25 5217  spanunsn 5482

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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