Theorem mpbir2and 888

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Hypotheses

Ref	Expression
mpbir2and.1	⊢ (φ → χ)
mpbir2and.2	⊢ (φ → θ)
mpbir2and.3	⊢ (φ → (ψ ↔ (χ ∧ θ)))

Assertion

Ref	Expression
mpbir2and	⊢ (φ → ψ)

Proof of Theorem mpbir2and

Step	Hyp	Ref	Expression
1		mpbir2and.1	. . 3 ⊢ (φ → χ)
2		mpbir2and.2	. . 3 ⊢ (φ → θ)
3	1, 2	jca 518	. 2 ⊢ (φ → (χ ∧ θ))
4		mpbir2and.3	. 2 ⊢ (φ → (ψ ↔ (χ ∧ θ)))
5	3, 4	mpbird 223	1 ⊢ (φ → ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ 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-an 360

This theorem is used by:  fvopab4t  5386