Theorem mpbiran 942

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  944  unssdif  3399  unssin  3403  inssun  3404  invdif  3406  pwpwab  4005  exmidexmid  4230  opabm  4316  regexmidlem1  4570  elirr  4578  en2lp  4591  wessep  4615  peano5  4635  relop  4817  ssrnres  5113  funopab  5294  funcnv2  5319  funcnveq  5322  fnres  5377  idref  5806  rnoprab  6009  elixp  6773  djuf1olem  7128  lbfzo0  10274  expghmap  14239  txdis1cn  14598