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  8569  sornom  10282  leltadd  11725  divmul13  11945  absmax  15420  fzomaxdif  15434  dmatsgrp  22724  comppfsc  23761  iocopnst  25171  mumul  27420  lgsdir2  27569  branmfn  32589  chirredlem2  32875  chirredlem4  32877  icoreclin  38114  relowlssretop  38120  pibt2  38174  frinfm  38488  fzmul  38494  fdc  38498  rpnnen3  43876