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Mirrors > Home > ILE Home > Th. List > anbi2ci | GIF version |
Description: Variant of anbi2i 430 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
bi.aa | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
anbi2ci | ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi.aa | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | anbi1i 431 | . 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)) |
3 | ancom 253 | . 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓)) | |
4 | 2, 3 | bitri 173 | 1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: clabel 2163 ordpwsucss 4291 asymref 4710 |
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