Theorem syl3an2 620

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	⊢ ((φ ∧ ψ ∧ χ) → θ)
syl3an2.2	⊢ (τ → ψ)

Assertion

Ref	Expression
syl3an2	⊢ ((φ ∧ τ ∧ χ) → θ)

Proof of Theorem syl3an2

Step	Hyp	Ref	Expression
1		syl3an.1	. . . 4 ⊢ ((φ ∧ ψ ∧ χ) → θ)
2	1	3exp 611	. . 3 ⊢ (φ → (ψ → (χ → θ)))
3		syl3an2.2	. . 3 ⊢ (τ → ψ)
4	2, 3	syl5 22	. 2 ⊢ (φ → (τ → (χ → θ)))
5	4	3imp 608	1 ⊢ ((φ ∧ τ ∧ χ) → θ)

Syntax hints:   → wi 2   ∧ w3a 581

This theorem is referenced by:  syl3an2b 623  syl3an2br 626  nndi 3180  nnmass 3181  qbtwnre 4650  his2subt 5052  projlem26 5218  chlubt 5426  mdsymlem5 5780

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  df-3an 583

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