Theorem mpbi2and 949

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 306	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒))
4		mpbi2and.3	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝜃))
5	3, 4	mpbid 147	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  supisoti  7193  remim  11392  resqrtcl  11561  divalgmod  12459  oddpwdclemxy  12712  divnumden  12739  numdensq  12745  prmdivdiv  12780  4sqlem7  12928  ismgmid2  13434  mnd1  13509  iscmnd  13856  imasring  14048  subrg1  14216  topgele  14724  lmcn2  14975