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Mirrors > Home > MPE Home > Th. List > eqrelrdv2 | Structured version Visualization version GIF version |
Description: A version of eqrelrdv 5658. (Contributed by Rodolfo Medina, 10-Oct-2010.) |
Ref | Expression |
---|---|
eqrelrdv2.1 | ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
Ref | Expression |
---|---|
eqrelrdv2 | ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelrdv2.1 | . . . 4 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | 1 | alrimivv 1923 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
3 | eqrel 5651 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
4 | 2, 3 | syl5ibr 248 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝜑 → 𝐴 = 𝐵)) |
5 | 4 | imp 409 | 1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1529 = wceq 1531 ∈ wcel 2108 〈cop 4565 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-in 3941 df-ss 3950 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: xpiindi 5699 fliftcnv 7056 dmtpos 7896 ercnv 8302 fpwwe2lem9 10052 invsym2 17025 eqbrrdv2 35991 dibglbN 38294 diclspsn 38322 dih1 38414 dihglbcpreN 38428 dihmeetlem4preN 38434 rfovcnvf1od 40341 |
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