Theorem prth 429

Description: Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema.'

Assertion

Ref	Expression
prth	⊢ (((φ → ψ) ∧ (χ → θ)) → ((φ ∧ χ) → (ψ ∧ θ)))

Proof of Theorem prth

Step	Hyp	Ref	Expression
1		pm3.2 232	. . . . 5 ⊢ (ψ → (θ → (ψ ∧ θ)))
2	1	syl3d 26	. . . 4 ⊢ (ψ → ((χ → θ) → (χ → (ψ ∧ θ))))
3	2	syl3 18	. . 3 ⊢ ((φ → ψ) → (φ → ((χ → θ) → (χ → (ψ ∧ θ)))))
4	3	com23 32	. 2 ⊢ ((φ → ψ) → ((χ → θ) → (φ → (χ → (ψ ∧ θ)))))
5	4	imp4b 283	1 ⊢ (((φ → ψ) ∧ (χ → θ)) → ((φ ∧ χ) → (ψ ∧ θ)))

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  anim12d 431  mo 1020  ssxp 2487  tfrlem5 2953  climunii 4883  hlimcaui 5141  hlimunii 5143  spanun 5450

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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