Theorem mpbiran 949

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  951  unssdif  3460  unssin  3464  inssun  3465  invdif  3467  pwpwab  4085  exmidexmid  4315  opabm  4405  regexmidlem1  4662  elirr  4670  en2lp  4683  wessep  4707  peano5  4727  relop  4912  ssrnres  5212  funopab  5394  funcnv2  5423  funcnveq  5426  fnres  5482  idref  5937  rnoprab  6146  elixp  6955  djuf1olem  7359  lbfzo0  10546  expghmap  14887  txdis1cn  15275