Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > anbi12ci | GIF version |
Description: Variant of anbi12i 433 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
anbi12.1 | ⊢ (𝜑 ↔ 𝜓) |
anbi12.2 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
anbi12ci | ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜃 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anbi12.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | anbi12.2 | . . 3 ⊢ (𝜒 ↔ 𝜃) | |
3 | 1, 2 | anbi12i 433 | . 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)) |
4 | ancom 253 | . 2 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓)) | |
5 | 3, 4 | bitri 173 | 1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜃 ∧ 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: opelopabsbALT 3996 cnvpom 4860 f1cnvcnv 5100 |
Copyright terms: Public domain | W3C validator |