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Mirrors > Home > ILE Home > Th. List > eqrd | GIF version |
Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
Ref | Expression |
---|---|
eqrd.0 | ⊢ Ⅎ𝑥𝜑 |
eqrd.1 | ⊢ Ⅎ𝑥𝐴 |
eqrd.2 | ⊢ Ⅎ𝑥𝐵 |
eqrd.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Ref | Expression |
---|---|
eqrd | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrd.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | eqrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | eqrd.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | biimpd 144 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
6 | 1, 2, 3, 5 | ssrd 3160 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
7 | 4 | biimprd 158 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) |
8 | 1, 3, 2, 7 | ssrd 3160 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
9 | 6, 8 | eqssd 3172 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 Ⅎwnf 1460 ∈ wcel 2148 Ⅎwnfc 2306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-in 3135 df-ss 3142 |
This theorem is referenced by: dfss4st 3368 imasnopn 13459 |
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