Theorem pm5.74d 262

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		pm5.74 259	. 2 ⊢ ((𝜓 → (𝜒 ↔ 𝜃)) ↔ ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
3	1, 2	sylib 208	1 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))

Syntax hints:   → wi 4   ↔ wb 196

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 197

This theorem is referenced by:  imbi2d  330  imim21b  382  pm5.74da  723  cbval2  2279  cbvaldva  2281  dvelimdf  2335  sbied  2409  dfiin2g  4553  oneqmini  5776  tfindsg  7060  findsg  7093  brecop  7840  dom2lem  7995  indpi  9729  nn0ind-raph  11477  cncls2  21077  ismbl2  23295  voliunlem3  23320  mdbr2  29155  dmdbr2  29162  mdsl2i  29181  mdsl2bi  29182  sgn3da  30603  bj-cbval2v  32737  wl-dral1d  33318  wl-equsald  33325  cvlsupr3  34631  cdleme32fva  35725  cdlemk33N  36197  cdlemk34  36198  ralbidar  38649  tfis2d  42427