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  7571  ov3  7587  omeulem2  8598  domtriomlem  10527  nsmallnq  11069  bposlem1  27620  pntrsumbnd  27902  elntg2  29572  mptsnunlem  38370  poimirlem27  38674  refressn  39574  frege92  45064  nzss  45410  modelaxreplem1  46076  ormklocald  47994  setis  50974