Theorem sylanbr 283

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 132	. 2 ⊢ (𝜑 → 𝜓)
3		sylanbr.2	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4	2, 3	sylan 281	1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  syl2anbr  290  mosubt  2856  r19.2m  3444  funfvdm  5477  caovimo  5957  tfrlem7  6207  iinerm  6494  expclzaplem  10310  expgt0  10319  expge0  10322  expge1  10323  rplpwr  11704