Theorem mpbidi 241

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 229	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	sylcom 30	1 ⊢ (𝜃 → (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 206

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 207

This theorem is referenced by:  ralxfr2d  5353  ovmpt4g  7503  ov3  7519  omeulem2  8508  domtriomlem  10350  nsmallnq  10886  bposlem1  27249  pntrsumbnd  27531  elntg2  29007  mptsnunlem  37482  poimirlem27  37787  refressn  38645  frege92  44138  nzss  44500  modelaxreplem1  45161  ormklocald  47060  setis  49885