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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-issetw | Structured version Visualization version GIF version |
Description: The closest one can get to isset 3506 without using ax-ext 2793. See also vexw 2805. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3506 using eleq2i 2904 (which requires ax-ext 2793 and df-cleq 2814). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-issetw.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
bj-issetw | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-issetwt 34189 | . 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)) | |
2 | bj-issetw.1 | . 2 ⊢ 𝜑 | |
3 | 1, 2 | mpg 1794 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 df-clab 2800 df-clel 2893 |
This theorem is referenced by: (None) |
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