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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dfsbc | Structured version Visualization version GIF version | ||
| Description: Proof of df-sbc 3754 when taking bj-df-sb 37157 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bj-dfsbc | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-clab 2748 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
| 2 | sb6 2125 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
| 3 | 1, 2 | bitri 278 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| 4 | 3 | anbi2i 634 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
| 5 | 4 | exbii 1875 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
| 6 | dfclel 2845 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
| 7 | bj-df-sb 37157 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
| 8 | 5, 6, 7 | 3bitr4i 306 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∀wal 1565 = wceq 1567 ∃wex 1806 [wsb 2097 ∈ wcel 2149 {cab 2747 [wsbc 3753 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-12 2219 ax-ext 2741 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1570 df-ex 1807 df-nf 1811 df-sb 2098 df-clab 2748 df-cleq 2761 df-clel 2844 df-v 3465 df-sbc 3754 |
| This theorem is referenced by: (None) |
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