Theorem pm5.74d 261

Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)

Hypothesis

Ref	Expression
pm5.74d.1	⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))

Assertion

Ref	Expression
pm5.74d	⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))

Proof of Theorem pm5.74d

Step	Hyp	Ref	Expression
1		pm5.74d.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
2		pm5.74 258	. 2 ⊢ ((𝜓 → (𝜒 ↔ 𝜃)) ↔ ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
3	1, 2	sylib 207	1 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))

Syntax hints:   → wi 4   ↔ wb 195

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 196

This theorem is referenced by:  imbi2d  329  imim21b  381  pm5.74da  719  cbval2  2267  cbvaldva  2269  dvelimdf  2323  sbied  2397  dfiin2g  4489  oneqmini  5693  tfindsg  6952  findsg  6985  brecop  7727  dom2lem  7881  indpi  9608  nn0ind-raph  11353  cncls2  20887  ismbl2  23102  voliunlem3  23127  mdbr2  28539  dmdbr2  28546  mdsl2i  28565  mdsl2bi  28566  sgn3da  29930  bj-cbval2v  31924  wl-dral1d  32497  wl-equsald  32504  cvlsupr3  33649  cdleme32fva  34743  cdlemk33N  35215  cdlemk34  35216  ralbidar  37670  tfis2d  42225