Theorem mpbidi 244

Description: A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)

Hypotheses

Ref	Expression
mpbidi.min	⊢ (𝜃 → (𝜑 → 𝜓))
mpbidi.maj	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
mpbidi	⊢ (𝜃 → (𝜑 → 𝜒))

Proof of Theorem mpbidi

Step	Hyp	Ref	Expression
1		mpbidi.min	. 2 ⊢ (𝜃 → (𝜑 → 𝜓))
2		mpbidi.maj	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	biimpd 232	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	sylcom 31	1 ⊢ (𝜃 → (𝜑 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  ralxfr2d  5375  ovmpt4g  7560  ov3  7576  omeulem2  8570  domtriomlem  10444  nsmallnq  10986  bposlem1  27520  pntrsumbnd  27802  elntg2  29442  mptsnunlem  38092  poimirlem27  38396  refressn  39281  frege92  44795  nzss  45141  modelaxreplem1  45801  ormklocald  47704  setis  50624