Theorem sylanblc 588

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 585	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

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 396

This theorem is referenced by:  uniintsn  4915  xmulpnf1  12937  odd2np1  15978  eltg3i  22019  restntr  22241  cmpcld  22461  rnelfm  23012  ovolctb2  24561  iscgra  27074  isinag  27103  isleag  27112  iseqlg  27132  omlsilem  29665  noextendseq  33797  mblfinlem3  35743