Theorem mpbidi 242

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 230	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	sylcom 30	1 ⊢ (𝜃 → (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 207

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 208

This theorem is referenced by:  ralxfr2d  5339  ovmpt4g  7503  ov3  7519  omeulem2  8508  domtriomlem  10355  nsmallnq  10891  bposlem1  27265  pntrsumbnd  27547  elntg2  29072  mptsnunlem  37700  poimirlem27  38014  refressn  38900  frege92  44399  nzss  44761  modelaxreplem1  45422  ormklocald  47319  setis  50188