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))

Syntax hints:   → wi 4   ↔ wb 209

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 210

This theorem is referenced by:  ralxfr2d  5381  ovmpt4g  7557  ov3  7573  omeulem2  8564  domtriomlem  10421  nsmallnq  10957  bposlem1  27448  pntrsumbnd  27730  elntg2  29335  mptsnunlem  37984  poimirlem27  38298  refressn  39182  frege92  44681  nzss  45027  modelaxreplem1  45687  ormklocald  47590  setis  50476