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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elabd2ALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of elabd2 3670 bypassing elab6g 3669 (and using sbiedvw 2095 instead of the ∀𝑥(𝑥 = 𝑦 → 𝜓) idiom). (Contributed by BJ, 16-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-elabd2ALT.ex | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| bj-elabd2ALT.eq | ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓}) | 
| bj-elabd2ALT.is | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| bj-elabd2ALT | ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-elabd2ALT.ex | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
| 3 | bj-elabd2ALT.eq | . . . . . 6 ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓}) | |
| 4 | 3 | eqcomd 2743 | . . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐵) | 
| 5 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → {𝑥 ∣ 𝜓} = 𝐵) | 
| 6 | 2, 5 | eleq12d 2835 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ 𝐵)) | 
| 7 | eqeq1 2741 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
| 8 | 7 | biimparc 479 | . . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝐴) | 
| 9 | 8 | anim2i 617 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝑦)) → (𝜑 ∧ 𝑥 = 𝐴)) | 
| 10 | 9 | anassrs 467 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → (𝜑 ∧ 𝑥 = 𝐴)) | 
| 11 | bj-elabd2ALT.is | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
| 12 | 10, 11 | syl 17 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | 
| 13 | 12 | sbiedvw 2095 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | 
| 14 | 6, 13 | bibi12d 345 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) ↔ (𝐴 ∈ 𝐵 ↔ 𝜒))) | 
| 15 | df-clab 2715 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
| 16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)) | 
| 17 | 1, 14, 16 | vtocld 3561 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 [wsb 2064 ∈ wcel 2108 {cab 2714 | 
| 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-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 | 
| This theorem is referenced by: (None) | 
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