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Mirrors > Home > HSE Home > Th. List > issubgoilem | Structured version Visualization version GIF version |
Description: Lemma for hhssabloilem 29040. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
issubgoilem.1 | ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦)) |
Ref | Expression |
---|---|
issubgoilem | ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7165 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦)) | |
2 | oveq1 7165 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦)) | |
3 | 1, 2 | eqeq12d 2839 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑥𝐺𝑦) ↔ (𝐴𝐻𝑦) = (𝐴𝐺𝑦))) |
4 | oveq2 7166 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵)) | |
5 | oveq2 7166 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵)) | |
6 | 4, 5 | eqeq12d 2839 | . 2 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝐴𝐺𝑦) ↔ (𝐴𝐻𝐵) = (𝐴𝐺𝐵))) |
7 | issubgoilem.1 | . 2 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦)) | |
8 | 3, 6, 7 | vtocl2ga 3577 | 1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: (None) |
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