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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elabd2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of elabd2 3592 bypassing elab6g 3591 (and using sbiedvw 2101 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 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
3 | bj-elabd2ALT.eq | . . . . . 6 ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓}) | |
4 | 3 | eqcomd 2744 | . . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐵) |
5 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → {𝑥 ∣ 𝜓} = 𝐵) |
6 | 2, 5 | eleq12d 2833 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ 𝐵)) |
7 | eqeq1 2742 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
8 | 7 | biimparc 483 | . . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝐴) |
9 | 8 | anim2i 620 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝑦)) → (𝜑 ∧ 𝑥 = 𝐴)) |
10 | 9 | anassrs 471 | . . . . 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 2101 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
14 | 6, 13 | bibi12d 349 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) ↔ (𝐴 ∈ 𝐵 ↔ 𝜒))) |
15 | df-clab 2716 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)) |
17 | 1, 14, 16 | vtocld 3483 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 [wsb 2071 ∈ wcel 2111 {cab 2715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 |
This theorem is referenced by: (None) |
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