Theorem mpbiran2 950

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  3015  ss0b  3547  eusv1  4572  eusv2nf  4576  eusv2  4577  opthprc  4800  opelres  5042  f1cnvcnv  5583  fores  5599  f1orn  5623  funfvdm  5739  dfoprab2  6099  tpostpos  6494  opelreal  8141  elreal2  8144  eqresr  8150  axprecex  8194  zeoxor  12551  isprm2  12810  toptopon  14875  bdeq0  16629  subctctexmid  16766