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Mirrors > Home > MPE Home > Th. List > elabd3 | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab 3661. (Contributed by Gino Giotto, 12-Oct-2024.) |
Ref | Expression |
---|---|
elabd3.ex | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elabd3.is | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
elabd3 | ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabd3.ex | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | eqidd 2725 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜓}) | |
3 | elabd3.is | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | elabd2 3653 | 1 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 |
This theorem is referenced by: sbcied 3815 |
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