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Mirrors > Home > ILE Home > Th. List > eqrelrdv2 | GIF version |
Description: A version of eqrelrdv 4379. (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 1752 | . . 3 ⊢ (φ → ∀x∀y(〈x, y〉 ∈ A ↔ 〈x, y〉 ∈ B)) |
3 | 2 | adantl 262 | . 2 ⊢ (((Rel A ∧ Rel B) ∧ φ) → ∀x∀y(〈x, y〉 ∈ A ↔ 〈x, y〉 ∈ B)) |
4 | eqrel 4372 | . . 3 ⊢ ((Rel A ∧ Rel B) → (A = B ↔ ∀x∀y(〈x, y〉 ∈ A ↔ 〈x, y〉 ∈ B))) | |
5 | 4 | adantr 261 | . 2 ⊢ (((Rel A ∧ Rel B) ∧ φ) → (A = B ↔ ∀x∀y(〈x, y〉 ∈ A ↔ 〈x, y〉 ∈ B))) |
6 | 3, 5 | mpbird 156 | 1 ⊢ (((Rel A ∧ Rel B) ∧ φ) → A = B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 = wceq 1242 ∈ wcel 1390 〈cop 3370 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 df-rel 4295 |
This theorem is referenced by: xpiindim 4416 fliftcnv 5378 dmtpos 5812 ercnv 6063 |
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