Theorem ancom1s 666

Description: Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Hypothesis

Ref	Expression
an32s.1	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
ancom1s	⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) → 𝜃)

Proof of Theorem ancom1s

Step	Hyp	Ref	Expression
1		pm3.22 465	. 2 ⊢ ((𝜓 ∧ 𝜑) → (𝜑 ∧ 𝜓))
2		an32s.1	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
3	1, 2	sylan 592	1 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  odi  8570  sornom  10283  leltadd  11726  divmul13  11946  absmax  15421  fzomaxdif  15435  dmatsgrp  22727  comppfsc  23764  iocopnst  25174  mumul  27425  lgsdir2  27574  branmfn  32594  chirredlem2  32880  chirredlem4  32882  icoreclin  38119  relowlssretop  38125  pibt2  38179  frinfm  38493  fzmul  38499  fdc  38503  rpnnen3  43881