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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj956 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj956.1 | ⊢ (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵) |
Ref | Expression |
---|---|
bnj956 | ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj956.1 | . . . 4 ⊢ (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵) | |
2 | eleq2 2677 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
3 | 2 | anbi1d 737 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
4 | 3 | alimi 1730 | . . . . . 6 ⊢ (∀𝑥 𝐴 = 𝐵 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
5 | exbi 1762 | . . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (∀𝑥 𝐴 = 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
7 | df-rex 2902 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) | |
8 | df-rex 2902 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
9 | 6, 7, 8 | 3bitr4g 302 | . . . 4 ⊢ (∀𝑥 𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) |
10 | 1, 9 | syl 17 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) |
11 | 10 | abbidv 2728 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}) |
12 | df-iun 4457 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} | |
13 | df-iun 4457 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶} | |
14 | 11, 12, 13 | 3eqtr4g 2669 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ∃wrex 2897 ∪ ciun 4455 |
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-rex 2902 df-iun 4457 |
This theorem is referenced by: bnj1316 30145 bnj953 30263 bnj1000 30265 bnj966 30268 |
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