Theorem sylanbr 285

Description: A syllogism inference. (Contributed by NM, 18-May-1994.)

Hypotheses

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

Assertion

Ref	Expression
sylanbr	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Proof of Theorem sylanbr

Step	Hyp	Ref	Expression
1		sylanbr.1	. . 3 ⊢ (𝜓 ↔ 𝜑)
2	1	biimpri 133	. 2 ⊢ (𝜑 → 𝜓)
3		sylanbr.2	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4	2, 3	sylan 283	1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  syl2anbr  292  mosubt  2915  r19.2m  3510  funfvdm  5580  caovimo  6068  tfrlem7  6318  iinerm  6607  expclzaplem  10544  expgt0  10553  expge0  10556  expge1  10557  rplpwr  12028