Theorem mpjaodan 761

Description: Eliminate a disjunction in a deduction. A translation of natural deduction rule ∨ E ( ∨ elimination), see natded in set.mm. (Contributed by Mario Carneiro, 29-May-2016.)

Hypotheses

Ref	Expression
jaodan.1	⊢ ((φ ∧ ψ) → χ)
jaodan.2	⊢ ((φ ∧ θ) → χ)
jaodan.3	⊢ (φ → (ψ ∨ θ))

Assertion

Ref	Expression
mpjaodan	⊢ (φ → χ)

Proof of Theorem mpjaodan

Step	Hyp	Ref	Expression
1		jaodan.3	. 2 ⊢ (φ → (ψ ∨ θ))
2		jaodan.1	. . 3 ⊢ ((φ ∧ ψ) → χ)
3		jaodan.2	. . 3 ⊢ ((φ ∧ θ) → χ)
4	2, 3	jaodan 760	. 2 ⊢ ((φ ∧ (ψ ∨ θ)) → χ)
5	1, 4	mpdan 649	1 ⊢ (φ → χ)

This proof depends on syntax axioms:   → wi 4   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by: (None)