Theorem ancom1s 653

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 580	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  8504  sornom  10190  leltadd  11622  divmul13  11845  absmax  15255  fzomaxdif  15269  dmatsgrp  22402  comppfsc  23435  iocopnst  24853  mumul  27107  lgsdir2  27257  branmfn  32067  chirredlem2  32353  chirredlem4  32355  icoreclin  37330  relowlssretop  37336  pibt2  37390  frinfm  37714  fzmul  37720  fdc  37724  rpnnen3  43005