Theorem mpbi2and 887

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
mpbi2and.1	⊢ (φ → ψ)
mpbi2and.2	⊢ (φ → χ)
mpbi2and.3	⊢ (φ → ((ψ ∧ χ) ↔ θ))

Assertion

Ref	Expression
mpbi2and	⊢ (φ → θ)

Proof of Theorem mpbi2and

Step	Hyp	Ref	Expression
1		mpbi2and.1	. . 3 ⊢ (φ → ψ)
2		mpbi2and.2	. . 3 ⊢ (φ → χ)
3	1, 2	jca 518	. 2 ⊢ (φ → (ψ ∧ χ))
4		mpbi2and.3	. 2 ⊢ (φ → ((ψ ∧ χ) ↔ θ))
5	3, 4	mpbid 201	1 ⊢ (φ → θ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ 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-an 360

This theorem is referenced by: (None)