Theorem pm3.48 430

Description: Theorem *3.48 of [WhiteheadRussell] p. 114.

Assertion

Ref	Expression
pm3.48	⊢ (((φ → ψ) ∧ (χ → θ)) → ((φ ∨ χ) → (ψ ∨ θ)))

Proof of Theorem pm3.48

Step	Hyp	Ref	Expression
1		pm3.26 256	. . . 4 ⊢ (((φ → ψ) ∧ (χ → θ)) → (φ → ψ))
2	1	con3d 87	. . 3 ⊢ (((φ → ψ) ∧ (χ → θ)) → (¬ ψ → ¬ φ))
3		pm3.27 260	. . 3 ⊢ (((φ → ψ) ∧ (χ → θ)) → (χ → θ))
4	2, 3	syl34d 29	. 2 ⊢ (((φ → ψ) ∧ (χ → θ)) → ((¬ φ → χ) → (¬ ψ → θ)))
5		df-or 197	. 2 ⊢ ((φ ∨ χ) ↔ (¬ φ → χ))
6		df-or 197	. 2 ⊢ ((ψ ∨ θ) ↔ (¬ ψ → θ))
7	4, 5, 6	3imtr4g 426	1 ⊢ (((φ → ψ) ∧ (χ → θ)) → ((φ ∨ χ) → (ψ ∨ θ)))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  orim12d 436  tz7.48lem 2993

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-or 197  df-an 198

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