Theorem mpbiran 946

Description: Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)

Hypotheses

Ref	Expression
mpbiran.1	⊢ 𝜓
mpbiran.2	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
mpbiran	⊢ (𝜑 ↔ 𝜒)

Proof of Theorem mpbiran

Step	Hyp	Ref	Expression
1		mpbiran.2	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2		mpbiran.1	. . 3 ⊢ 𝜓
3	2	biantrur 303	. 2 ⊢ (𝜒 ↔ (𝜓 ∧ 𝜒))
4	1, 3	bitr4i 187	1 ⊢ (𝜑 ↔ 𝜒)

Syntax hints:   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mpbir2an  948  unssdif  3439  unssin  3443  inssun  3444  invdif  3446  pwpwab  4053  exmidexmid  4281  opabm  4370  regexmidlem1  4626  elirr  4634  en2lp  4647  wessep  4671  peano5  4691  relop  4875  ssrnres  5174  funopab  5356  funcnv2  5384  funcnveq  5387  fnres  5443  idref  5889  rnoprab  6096  elixp  6865  djuf1olem  7236  lbfzo0  10398  expghmap  14592  txdis1cn  14973