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Mirrors > Home > MPE Home > Th. List > elabgw | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, using two substitution hypotheses to avoid a disjoint variable condition on 𝑥 and 𝐴. This is to elabg 3676 what sbievw2 2095 is to sbievw 2090. (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 2826 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
2 | elabgw.2 | . 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | df-clab 2712 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | elabgw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 4 | sbievw 2090 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 3, 5 | bitri 275 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
7 | 1, 2, 6 | vtoclbg 3556 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 [wsb 2061 ∈ wcel 2105 {cab 2711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 |
This theorem is referenced by: elab2gw 3679 |
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