Theorem mpbiran2 941

Description: Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)

Hypotheses

Ref	Expression
mpbiran2.1	⊢ 𝜒
mpbiran2.2	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
mpbiran2	⊢ (𝜑 ↔ 𝜓)

Proof of Theorem mpbiran2

Step	Hyp	Ref	Expression
1		mpbiran2.2	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2		mpbiran2.1	. . 3 ⊢ 𝜒
3	2	biantru 302	. 2 ⊢ (𝜓 ↔ (𝜓 ∧ 𝜒))
4	1, 3	bitr4i 187	1 ⊢ (𝜑 ↔ 𝜓)

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:  reueq  2934  ss0b  3460  eusv1  4446  eusv2nf  4450  eusv2  4451  opthprc  4671  opelres  4905  f1cnvcnv  5424  fores  5439  f1orn  5463  funfvdm  5571  dfoprab2  5912  tpostpos  6255  opelreal  7801  elreal2  7804  eqresr  7810  axprecex  7854  zeoxor  11839  isprm2  12082  toptopon  13085  bdeq0  14177  subctctexmid  14309