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Mirrors > Home > MPE Home > Th. List > rabsnt | Structured version Visualization version GIF version |
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
rabsnt.1 | ⊢ 𝐵 ∈ V |
rabsnt.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rabsnt | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabsnt.1 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | 1 | snid 4627 | . . 3 ⊢ 𝐵 ∈ {𝐵} |
3 | id 22 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵}) | |
4 | 2, 3 | eleqtrrid 2845 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
5 | rabsnt.2 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
6 | 5 | elrab 3650 | . . 3 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ∈ 𝐴 ∧ 𝜓)) |
7 | 6 | simprbi 498 | . 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜓) |
8 | 4, 7 | syl 17 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {crab 3410 Vcvv 3448 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-sn 4592 |
This theorem is referenced by: ddemeas 32875 |
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