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| Mirrors > Home > MPE Home > Th. List > iseqsetv-clel | Structured version Visualization version GIF version | ||
| Description: Alternate proof of iseqsetv-cleq 2805. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. Use this theorem in contexts where df-cleq 2728 or ax-ext 2707 is not referenced elsewhere in your proof. It is proven from a specific implementation (class builder, axiom df-clab 2714) of the primitive term 𝑥 ∈ 𝐴. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| iseqsetv-clel | ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issettru 2818 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤}) | |
| 2 | issettru 2818 | . 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤}) | |
| 3 | 1, 2 | bitr4i 278 | 1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ⊤wtru 1541 ∃wex 1779 ∈ wcel 2108 {cab 2713 |
| 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-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-clel 2815 |
| This theorem is referenced by: elisset 2822 bj-issettruALTV 36852 bj-issetwt 36854 bj-vtoclg1f1 36896 |
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