Theorem sylanblc 592

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

Hypotheses

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

Assertion

Ref	Expression
sylanblc	⊢ (𝜑 → 𝜃)

Proof of Theorem sylanblc

Step	Hyp	Ref	Expression
1		sylanblc.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylanblc.2	. 2 ⊢ 𝜒
3		sylanblc.3	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃)
4	3	biimpi 219	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
5	1, 2, 4	sylancl 589	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

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 210  df-an 400

This theorem is referenced by:  uniintsn  4875  xmulpnf1  12655  odd2np1  15682  eltg3i  21566  restntr  21787  cmpcld  22007  rnelfm  22558  ovolctb2  24096  iscgra  26603  isinag  26632  isleag  26641  iseqlg  26661  omlsilem  29185  noextendseq  33287  mblfinlem3  35096