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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 2142, ax-11 2158, ax-12 2175. It also avoids a disjoint variable condition on 𝑥 and 𝐴. This is to elabg 3614 what sbievw2 2104 is to sbievw 2100. (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 2877 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
2 | elabgw.2 | . 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | df-clab 2777 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | elabgw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 4 | sbievw 2100 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 3, 5 | bitri 278 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
7 | 1, 2, 6 | vtoclbg 3517 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 [wsb 2069 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: elab2gw 3613 elgch 10033 |
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