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Mirrors > Home > ILE Home > Th. List > an12 | GIF version |
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.) |
Ref | Expression |
---|---|
an12 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒))) |
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))) |
Colors of variables: wff set class |
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 2851 rmoan 2921 2reuswapdc 2925 reuind 2926 2rmorex 2927 sbccomlem 3020 elunirab 3796 rexxfrd 4435 opeliunxp 4653 elres 4914 resoprab 5929 ov6g 5970 opabex3d 6081 opabex3 6082 xpassen 6787 distrnqg 7319 distrnq0 7391 rexuz2 9510 2clim 11228 isbasis2g 12590 tgval2 12598 |
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