Theorem jaao 495

Description: Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)

Hypotheses

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

Assertion

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

Proof of Theorem jaao

Step	Hyp	Ref	Expression
1		jaao.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	adantr 451	. 2 ⊢ ((φ ∧ θ) → (ψ → χ))
3		jaao.2	. . 3 ⊢ (θ → (τ → χ))
4	3	adantl 452	. 2 ⊢ ((φ ∧ θ) → (τ → χ))
5	2, 4	jaod 369	1 ⊢ ((φ ∧ θ) → ((ψ ∨ τ) → 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:  pm3.44  497  pm3.48  806  prlem1  928  funun  5147