Theorem mpbir3an 1134

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.)

Hypotheses

Ref	Expression
mpbir3an.1	⊢ ψ
mpbir3an.2	⊢ χ
mpbir3an.3	⊢ θ
mpbir3an.4	⊢ (φ ↔ (ψ ∧ χ ∧ θ))

Assertion

Ref	Expression
mpbir3an	⊢ φ

Proof of Theorem mpbir3an

Step	Hyp	Ref	Expression
1		mpbir3an.1	. . 3 ⊢ ψ
2		mpbir3an.2	. . 3 ⊢ χ
3		mpbir3an.3	. . 3 ⊢ θ
4	1, 2, 3	3pm3.2i 1130	. 2 ⊢ (ψ ∧ χ ∧ θ)
5		mpbir3an.4	. 2 ⊢ (φ ↔ (ψ ∧ χ ∧ θ))
6	4, 5	mpbir 200	1 ⊢ φ

This proof depends on syntax axioms:   ↔ wb 176   ∧ w3a 934

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  df-3an 936

This theorem is used by:  sfin01  4529  pw1fnf1o  5856