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  5372  ovmpt4g  7567  ov3  7583  omeulem2  8591  domtriomlem  10520  nsmallnq  11062  bposlem1  27611  pntrsumbnd  27893  elntg2  29563  mptsnunlem  38261  poimirlem27  38565  refressn  39465  frege92  44954  nzss  45300  modelaxreplem1  45967  ormklocald  47885  setis  50790