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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 3497 without using ax-ext 2705. See also vexw 2717. Note that the only disjoint variable condition is between 𝑦 and 𝐴. From there, one can prove isset 3497 using eleq2i 2830 (which requires ax-ext 2705 and df-cleq 2726). (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 36790 | . 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴)) | |
2 | bj-issetw.1 | . 2 ⊢ 𝜑 | |
3 | 1, 2 | mpg 1795 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∃wex 1777 ∈ wcel 2103 {cab 2711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2712 df-clel 2813 |
This theorem is referenced by: (None) |
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