Theorem mpbiran 940

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  942  unssdif  3370  unssin  3374  inssun  3375  invdif  3377  pwpwab  3974  exmidexmid  4196  opabm  4280  regexmidlem1  4532  elirr  4540  en2lp  4553  wessep  4577  peano5  4597  relop  4777  ssrnres  5071  funopab  5251  funcnv2  5276  funcnveq  5279  fnres  5332  idref  5757  rnoprab  5957  elixp  6704  djuf1olem  7051  lbfzo0  10180  txdis1cn  13748