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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 3611. (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 2741 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜓}) | |
3 | elabd3.is | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | elabd2 3603 | 1 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {cab 2717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 |
This theorem is referenced by: sbcied 3765 |
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