Theorem sylanblc 589

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 215	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
5	1, 2, 4	sylancl 586	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

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 206  df-an 397

This theorem is referenced by:  uniintsn  4918  xmulpnf1  13008  odd2np1  16050  eltg3i  22111  restntr  22333  cmpcld  22553  rnelfm  23104  ovolctb2  24656  iscgra  27170  isinag  27199  isleag  27208  iseqlg  27228  omlsilem  29764  noextendseq  33870  mblfinlem3  35816