| Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dfsbc | Structured version Visualization version GIF version | ||
| Description: Proof of df-sbc 3744 when taking bj-df-sb 37238 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 2740 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
| 2 | sb6 2117 | . . . . 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 1876 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
| 6 | dfclel 2837 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
| 7 | bj-df-sb 37238 | . 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 1566 = wceq 1568 ∃wex 1807 [wsb 2094 ∈ wcel 2141 {cab 2739 [wsbc 3743 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-12 2211 ax-ext 2733 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1571 df-ex 1808 df-nf 1812 df-sb 2095 df-clab 2740 df-cleq 2753 df-clel 2836 df-v 3455 df-sbc 3744 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |