Theorem mpbir3and 1135

Description: Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)

Hypotheses

Ref	Expression
mpbir3and.1	⊢ (φ → χ)
mpbir3and.2	⊢ (φ → θ)
mpbir3and.3	⊢ (φ → τ)
mpbir3and.4	⊢ (φ → (ψ ↔ (χ ∧ θ ∧ τ)))

Assertion

Ref	Expression
mpbir3and	⊢ (φ → ψ)

Proof of Theorem mpbir3and

Step	Hyp	Ref	Expression
1		mpbir3and.1	. . 3 ⊢ (φ → χ)
2		mpbir3and.2	. . 3 ⊢ (φ → θ)
3		mpbir3and.3	. . 3 ⊢ (φ → τ)
4	1, 2, 3	3jca 1132	. 2 ⊢ (φ → (χ ∧ θ ∧ τ))
5		mpbir3and.4	. 2 ⊢ (φ → (ψ ↔ (χ ∧ θ ∧ τ)))
6	4, 5	mpbird 223	1 ⊢ (φ → ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934

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  df-3an 936

This theorem is referenced by:  spaccl  6287