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Mirrors > Home > ILE Home > Th. List > bibi12i | GIF version |
Description: The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
bibi.a | ⊢ (𝜑 ↔ 𝜓) |
bibi12.2 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
bibi12i | ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bibi12.2 | . . 3 ⊢ (𝜒 ↔ 𝜃) | |
2 | 1 | bibi2i 227 | . 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜃)) |
3 | bibi.a | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
4 | 3 | bibi1i 228 | . 2 ⊢ ((𝜑 ↔ 𝜃) ↔ (𝜓 ↔ 𝜃)) |
5 | 2, 4 | bitri 184 | 1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: pm5.7dc 954 asymref 5009 rexrnmpt 5654 uzennn 10409 |
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