Theorem sylanr2 409

Description: A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Hypotheses

Ref	Expression
sylanr2.1	⊢ (𝜑 → 𝜃)
sylanr2.2	⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)

Assertion

Ref	Expression
sylanr2	⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏)

Proof of Theorem sylanr2

Step	Hyp	Ref	Expression
1		sylanr2.1	. . 3 ⊢ (𝜑 → 𝜃)
2	1	anim2i 342	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜃))
3		sylanr2.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 286	1 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is used by:  adantrrl  490  adantrrr  491  1stconst  6457  2ndconst  6458  ltexprlemopl  7969  ltexprlemopu  7971  mulsub  8730  fzsubel  10477  expsubap  11039  tgcl  15256