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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 2718 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
| 2 | velsn 4642 | . . . . 5 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦) | |
| 3 | 1, 2 | bibi12i 339 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦)) | 
| 4 | biimpr 220 | . . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑)) | |
| 5 | 3, 4 | sylbi 217 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦 → 𝜑)) | 
| 6 | 5 | alimi 1811 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | 
| 7 | nfab1 2907 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
| 8 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥{𝑦} | |
| 9 | 7, 8 | cleqf 2934 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦})) | 
| 10 | sb6 2085 | . 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 206 ∀wal 1538 = wceq 1540 [wsb 2064 ∈ wcel 2108 {cab 2714 {csn 4626 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sn 4627 | 
| This theorem is referenced by: (None) | 
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