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Mirrors > Home > MPE Home > Th. List > elabgw | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, using implicit substitution and an intermediate setvar 𝑦 to avoid ax-10 2143, ax-11 2159, ax-12 2176. It also avoids a disjoint variable condition on 𝑥 and 𝐴. This is to elabg 3585 what sbievw2 2105 is to sbievw 2101. (Contributed by SN, 20-Apr-2024.) |
Ref | Expression |
---|---|
elabgw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
elabgw.2 | ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
elabgw | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2838 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
2 | elabgw.2 | . 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | df-clab 2737 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | elabgw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 4 | sbievw 2101 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 3, 5 | bitri 278 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
7 | 1, 2, 6 | vtoclbg 3485 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1539 [wsb 2070 ∈ wcel 2112 {cab 2736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 |
This theorem is referenced by: elab2gw 3584 elgch 10067 |
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