Theorem 3jaoi 633

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 603	. 2 ⊢ ((φ → ψ) ∧ (χ → ψ) ∧ (θ → ψ))
5		3jao 632	. 2 ⊢ (((φ → ψ) ∧ (χ → ψ) ∧ (θ → ψ)) → ((φ ∨ χ ∨ θ) → ψ))
6	4, 5	ax-mp 6	1 ⊢ ((φ ∨ χ ∨ θ) → ψ)

Syntax hints:   → wi 2   ∨ w3o 580   ∧ w3a 581

This theorem is referenced by:  ordzsl 2366  oawordeulem 3156  r1val1 3502  rankr1 3518  znegclt 4588

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  df-3or 582  df-3an 583

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