Theorem sylan2d 606

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Hypotheses

Ref	Expression
sylan2d.1	⊢ (𝜑 → (𝜓 → 𝜒))
sylan2d.2	⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏))

Assertion

Ref	Expression
sylan2d	⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏))

Proof of Theorem sylan2d

Step	Hyp	Ref	Expression
1		sylan2d.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		sylan2d.2	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏))
3	2	ancomsd 465	. . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))
4	1, 3	syland 604	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏))
5	4	ancomsd 465	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏))

Syntax hints:   → wi 4   ∧ 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 207  df-an 396

This theorem is referenced by:  sylan2i  607  syl2and  609  swopo  5550  fprlem1  8250  unblem1  9202  frrlem15  9681  prodgt02  12003  lo1mul  15590  infpnlem1  16881  ghmcnp  24080  ulmcaulem  26359  ulmcau  26360  shintcli  31400  ballotlemfc0  34637  ballotlemfcc  34638  btwnxfr  36238  endofsegid  36267  bj-bary1lem1  37625  matunitlindflem1  37937  ltcvrntr  39870  poml4N  40399