Theorem an12 551

Description: Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)

Assertion

Ref	Expression
an12	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))

Proof of Theorem an12

Step	Hyp	Ref	Expression
1		ancom 264	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2	1	anbi1i 454	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒))
3		anass 399	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
4		anass 399	. 2 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
5	2, 3, 4	3bitr3i 209	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))

Syntax hints:   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  an32  552  an13  553  an12s  555  an4  576  ceqsrexv  2856  rmoan  2926  2reuswapdc  2930  reuind  2931  2rmorex  2932  sbccomlem  3025  elunirab  3802  rexxfrd  4441  opeliunxp  4659  elres  4920  resoprab  5938  ov6g  5979  opabex3d  6089  opabex3  6090  xpassen  6796  distrnqg  7328  distrnq0  7400  rexuz2  9519  2clim  11242  isbasis2g  12683  tgval2  12691