Theorem ancom1s 654

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		pm3.22 459	. 2 ⊢ ((𝜓 ∧ 𝜑) → (𝜑 ∧ 𝜓))
2		an32s.1	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
3	1, 2	sylan 581	1 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) → 𝜃)

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:  odi  8516  sornom  10199  leltadd  11633  divmul13  11856  absmax  15265  fzomaxdif  15279  dmatsgrp  22455  comppfsc  23488  iocopnst  24905  mumul  27159  lgsdir2  27309  branmfn  32192  chirredlem2  32478  chirredlem4  32480  icoreclin  37606  relowlssretop  37612  pibt2  37666  frinfm  37980  fzmul  37986  fdc  37990  rpnnen3  43383