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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 3667. (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 2729 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜓}) | |
3 | elabd3.is | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | elabd2 3658 | 1 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 |
This theorem is referenced by: sbcied 3822 |
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