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| Mirrors > Home > MPE Home > Th. List > iseqsetv-cleq | Structured version Visualization version GIF version | ||
| Description: Alternate proof of iseqsetv-clel 2819. The expression ∃𝑥𝑥 = 𝐴 does
not depend on a particular choice of the set variable. The proof here
avoids df-clab 2714, df-clel 2815 and ax-8 2110, but instead is based on
ax-9 2118, ax-ext 2707 and df-cleq 2728. In particular it still accepts
𝑥
∈ 𝐴 being a
primitive syntax term, not assuming any specific
semantics (like elementhood in some form).
Use it in contexts where you want to avoid df-clab 2714, or you need df-cleq 2728 anyway. See the alternative version , not using df-cleq 2728 or ax-ext 2707 or ax-9 2118. (Contributed by Wolf Lammen, 6-Aug-2025.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| iseqsetv-cleq | ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iseqsetvlem 2804 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴) | |
| 2 | iseqsetvlem 2804 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴) | |
| 3 | 1, 2 | bitr4i 278 | 1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2728 |
| This theorem is referenced by: clelab 2886 |
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