Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqrelrdv2 | Structured version Visualization version GIF version |
Description: A version of eqrelrdv 5139. (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 1843 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
3 | eqrel 5132 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
4 | 2, 3 | syl5ibr 235 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝜑 → 𝐴 = 𝐵)) |
5 | 4 | imp 444 | 1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 〈cop 4131 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: xpiindi 5179 fliftcnv 6461 dmtpos 7251 ercnv 7650 fpwwe2lem9 9339 invsym2 16246 eqbrrdv2 33166 dibglbN 35473 diclspsn 35501 dih1 35593 dihglbcpreN 35607 dihmeetlem4preN 35613 rfovcnvf1od 37318 |
Copyright terms: Public domain | W3C validator |