Theorem sylanblrc 697

Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)

Hypotheses

Ref	Expression
sylanblrc.1	⊢ (𝜑 → 𝜓)
sylanblrc.2	⊢ 𝜒
sylanblrc.3	⊢ (𝜃 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
sylanblrc	⊢ (𝜑 → 𝜃)

Proof of Theorem sylanblrc

Step	Hyp	Ref	Expression
1		sylanblrc.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylanblrc.2	. 2 ⊢ 𝜒
3		sylanblrc.3	. . 3 ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒))
4	3	biimpri 218	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
5	1, 2, 4	sylancl 694	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  fnwelem  7292  tfrlem10  7483  gruina  9640  dfac14  21421  1trld  27002  1stmbfm  30322  2ndmbfm  30323  bj-projval  32984  rfcnpre1  39178