Theorem biancomd 463

Description: Commuting conjunction in a biconditional, deduction form. (Contributed by Peter Mazsa, 3-Oct-2018.)

Hypothesis

Ref	Expression
biancomd.1	⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒)))

Assertion

Ref	Expression
biancomd	⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))

Proof of Theorem biancomd

Step	Hyp	Ref	Expression
1		biancomd.1	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒)))
2		ancom 460	. 2 ⊢ ((𝜃 ∧ 𝜒) ↔ (𝜒 ∧ 𝜃))
3	1, 2	bitrdi 287	1 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396

This theorem is referenced by:  ibar  528  rbaibd  540  pm4.71rd  562  anbi1cd  633  mpbiran2d  705  naddcom  8684  elpmg  8840  letri3  11304  mulsuble0b  12091  xrletri3  13138  qbtwnre  13183  iooneg  13453  invsym  17714  subsubc  17808  lsslss  20717  znleval  21330  restopn2  22902  elflim2  23689  ismet2  24060  mbfi1fseqlem4  25469  deg1ldg  25846  sincosq1sgn  26245  lgsquadlem3  27122  numclwwlkqhash  29896  rmounid  32003  dfrdg4  35228  bj-19.41t  35956  bj-0int  36286  orddif0suc  42321  dflim7  42326  naddsuc2  42446  mpbiran4d  47571