Theorem jaoa 496

Description: Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)

Hypotheses

Ref	Expression
jaao.1	⊢ (φ → (ψ → χ))
jaao.2	⊢ (θ → (τ → χ))

Assertion

Ref	Expression
jaoa	⊢ ((φ ∨ θ) → ((ψ ∧ τ) → χ))

Proof of Theorem jaoa

Step	Hyp	Ref	Expression
1		jaao.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	adantrd 454	. 2 ⊢ (φ → ((ψ ∧ τ) → χ))
3		jaao.2	. . 3 ⊢ (θ → (τ → χ))
4	3	adantld 453	. 2 ⊢ (θ → ((ψ ∧ τ) → χ))
5	2, 4	jaoi 368	1 ⊢ ((φ ∨ θ) → ((ψ ∧ τ) → χ))

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by:  pm4.79  566