Theorem pm5.74d 264

Description: Distribution of implication over biconditional (deduction form). (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		pm5.74 261	. 2 ⊢ ((𝜓 → (𝜒 ↔ 𝜃)) ↔ ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
3	1, 2	sylib 209	1 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))

Syntax hints:   → wi 4   ↔ wb 197

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 198

This theorem is referenced by:  imbi2d  331  imim21b  383  pm5.74da  838  cbval2  2382  cbvaldva  2384  dvelimdf  2429  sbied  2500  dfiin2g  4709  oneqmini  5959  tfindsg  7258  findsg  7291  brecop  8043  dom2lem  8200  indpi  9982  nn0ind-raph  11724  cncls2  21357  ismbl2  23585  voliunlem3  23610  mdbr2  29611  dmdbr2  29618  mdsl2i  29637  mdsl2bi  29638  sgn3da  31051  bj-cbval2v  33171  wl-dral1d  33743  wl-equsald  33750  cvlsupr3  35300  cdleme32fva  36393  cdlemk33N  36865  cdlemk34  36866  ralbidar  39321  tfis2d  43096