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Mirrors > Home > MPE Home > Th. List > Mathboxes > absnsb | Structured version Visualization version GIF version |
Description: If the class abstraction {𝑥 ∣ 𝜑} associated with the wff 𝜑 is a singleton, the wff is true for the singleton element. (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
absnsb | ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2719 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | velsn 4577 | . . . . 5 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦) | |
3 | 1, 2 | bibi12i 340 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦)) |
4 | biimpr 219 | . . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑)) | |
5 | 3, 4 | sylbi 216 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦 → 𝜑)) |
6 | 5 | alimi 1814 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
7 | nfab1 2909 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
8 | nfcv 2907 | . . 3 ⊢ Ⅎ𝑥{𝑦} | |
9 | 7, 8 | cleqf 2938 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦})) |
10 | sb6 2088 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
11 | 6, 9, 10 | 3imtr4i 292 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 = wceq 1539 [wsb 2067 ∈ wcel 2106 {cab 2715 {csn 4561 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-v 3434 df-sn 4562 |
This theorem is referenced by: (None) |
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