![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3681. (Contributed by GG, 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 2736 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜓}) | |
3 | elabd3.is | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | elabd2 3670 | 1 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: sbcied 3837 |
Copyright terms: Public domain | W3C validator |