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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elabd2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of elabd2 3659 bypassing elab6g 3658 (and using sbiedvw 2094 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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
3 | bj-elabd2ALT.eq | . . . . . 6 ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓}) | |
4 | 3 | eqcomd 2736 | . . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐵) |
5 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → {𝑥 ∣ 𝜓} = 𝐵) |
6 | 2, 5 | eleq12d 2825 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ 𝐵)) |
7 | eqeq1 2734 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
8 | 7 | biimparc 478 | . . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝐴) |
9 | 8 | anim2i 615 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝑦)) → (𝜑 ∧ 𝑥 = 𝐴)) |
10 | 9 | anassrs 466 | . . . . 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 2094 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
14 | 6, 13 | bibi12d 344 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) ↔ (𝐴 ∈ 𝐵 ↔ 𝜒))) |
15 | df-clab 2708 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)) |
17 | 1, 14, 16 | vtocld 3546 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 [wsb 2065 ∈ wcel 2104 {cab 2707 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 |
This theorem is referenced by: (None) |
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