Theorem 3jaoi 889

Description: Disjunction of 3 antecedents (inference).

Hypotheses

Ref	Expression
3jaoi.1	⊢ (φ → ψ)
3jaoi.2	⊢ (χ → ψ)
3jaoi.3	⊢ (θ → ψ)

Assertion

Ref	Expression
3jaoi	⊢ ((φ ⋁ χ ⋁ θ) → ψ)

Proof of Theorem 3jaoi

Step	Hyp	Ref	Expression
1		3jaoi.1	. . 3 ⊢ (φ → ψ)
2		3jaoi.2	. . 3 ⊢ (χ → ψ)
3		3jaoi.3	. . 3 ⊢ (θ → ψ)
4	1, 2, 3	3pm3.2i 820	. 2 ⊢ ((φ → ψ) ⋀ (χ → ψ) ⋀ (θ → ψ))
5		3jao 888	. 2 ⊢ (((φ → ψ) ⋀ (χ → ψ) ⋀ (θ → ψ)) → ((φ ⋁ χ ⋁ θ) → ψ))
6	4, 5	ax-mp 7	1 ⊢ ((φ ⋁ χ ⋁ θ) → ψ)

Syntax hints:   → wi 3   ⋁ w3o 776   ⋀ w3a 777

This theorem is referenced by:  3jaoian 891  ordzsl 3122  oawordeulem 4194  r1val1 4668  rankr1 4684  xrltnrt 5553  xrsupsslem 6078  xrinfmsslem 6079  znegclt 6165

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779

Copyright terms: Public domain