Theorem sylan2d 605

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 603	. 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  606  syl2and  608  swopo  5603  fprlem1  8325  wfrlem5OLD  8353  unblem1  9328  frrlem15  9797  prodgt02  12115  lo1mul  15664  infpnlem1  16948  ghmcnp  24123  ulmcaulem  26437  ulmcau  26438  shintcli  31348  ballotlemfc0  34495  ballotlemfcc  34496  btwnxfr  36057  endofsegid  36086  bj-bary1lem1  37312  matunitlindflem1  37623  ltcvrntr  39426  poml4N  39955