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:  mpt3fvot2d  7714  odi  8615  sornom  10382  leltadd  11827  divmul13  12047  absmax  15526  fzomaxdif  15540  dmatsgrp  22844  comppfsc  23881  iocopnst  25291  mumul  27538  lgsdir2  27687  branmfn  32737  chirredlem2  33023  chirredlem4  33025  icoreclin  38472  relowlssretop  38478  pibt2  38532  frinfm  38861  fzmul  38867  fdc  38871  rpnnen3  44232