Theorem mpbidi 243

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 231	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	sylcom 30	1 ⊢ (𝜃 → (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208

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 209

This theorem is referenced by:  ralxfr2d  5302  ovmpt4g  7291  ov3  7305  omeulem2  8203  domtriomlem  9858  nsmallnq  10393  bposlem1  25854  pntrsumbnd  26136  elntg2  26765  mptsnunlem  34613  poimirlem27  34913  frege92  40294  nzss  40642  setis  44794