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Mirrors > Home > ILE Home > Th. List > oplem1 | GIF version |
Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
Ref | Expression |
---|---|
oplem1.1 | ⊢ (𝜑 → (𝜓 ∨ 𝜒)) |
oplem1.2 | ⊢ (𝜑 → (𝜃 ∨ 𝜏)) |
oplem1.3 | ⊢ (𝜓 ↔ 𝜃) |
oplem1.4 | ⊢ (𝜒 → (𝜃 ↔ 𝜏)) |
Ref | Expression |
---|---|
oplem1 | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oplem1.1 | . 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒)) | |
2 | idd 21 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜓)) | |
3 | oplem1.2 | . . . . 5 ⊢ (𝜑 → (𝜃 ∨ 𝜏)) | |
4 | ax-1 6 | . . . . . 6 ⊢ (𝜃 → (𝜒 → 𝜃)) | |
5 | oplem1.4 | . . . . . . 7 ⊢ (𝜒 → (𝜃 ↔ 𝜏)) | |
6 | 5 | biimprcd 159 | . . . . . 6 ⊢ (𝜏 → (𝜒 → 𝜃)) |
7 | 4, 6 | jaoi 711 | . . . . 5 ⊢ ((𝜃 ∨ 𝜏) → (𝜒 → 𝜃)) |
8 | 3, 7 | syl 14 | . . . 4 ⊢ (𝜑 → (𝜒 → 𝜃)) |
9 | oplem1.3 | . . . 4 ⊢ (𝜓 ↔ 𝜃) | |
10 | 8, 9 | syl6ibr 161 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜓)) |
11 | 2, 10 | jaod 712 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) → 𝜓)) |
12 | 1, 11 | mpd 13 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 703 |
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 704 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: preqr1g 3751 preqr1 3753 |
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