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Mirrors > Home > ILE Home > Th. List > orbi1i | GIF version |
Description: Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
orbi2i.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
orbi1i | ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orcom 718 | . 2 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜒 ∨ 𝜑)) | |
2 | orbi2i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | orbi2i 752 | . 2 ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) |
4 | orcom 718 | . 2 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒)) | |
5 | 1, 3, 4 | 3bitri 205 | 1 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: orbi12i 754 orordi 763 dcan 919 3or6 1305 19.45 1663 sbequilem 1818 unass 3265 frecsuc 6356 elznn0nn 9186 |
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