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Mirrors > Home > ILE Home > Th. List > rabeqbidv | GIF version |
Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) |
Ref | Expression |
---|---|
rabeqbidv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
rabeqbidv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabeqbidv | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqbidv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | rabeq 2551 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
4 | rabeqbidv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 4 | rabbidv 2549 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
6 | 3, 5 | eqtrd 2072 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 {crab 2310 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 |
This theorem is referenced by: mpt2xopoveq 5855 |
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