Theorem sylan12 355

Description: A syllogism inference.

Hypotheses

Ref	Expression
sylan12.1	⊢ (((φ ∧ ψ) ∧ χ) → θ)
sylan12.2	⊢ (τ → ψ)

Assertion

Ref	Expression
sylan12	⊢ (((φ ∧ τ) ∧ χ) → θ)

Proof of Theorem sylan12

Step	Hyp	Ref	Expression
1		sylan12.1	. 2 ⊢ (((φ ∧ ψ) ∧ χ) → θ)
2		sylan12.2	. . 3 ⊢ (τ → ψ)
3	2	anim2i 270	. 2 ⊢ ((φ ∧ τ) → (φ ∧ ψ))
4	1, 3	sylan 343	1 ⊢ (((φ ∧ τ) ∧ χ) → θ)

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  fvco3 2867  oesuc 3134  oelim 3137  divadddivt 4264  infxpidmlem12 4944

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

metamath.org