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Mirrors > Home > MPE Home > Th. List > iseqsetv-clel | Structured version Visualization version GIF version |
Description: Alternate proof of iseqsetv-cleq 2802. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. Use this theorem in contexts where df-cleq 2725 or ax-ext 2704 is not referenced elsewhere in your proof. It is proven from a specific implementation (class builder, axiom df-clab 2711) 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 2815 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑧 ∣ ⊤}) | |
2 | issettru 2815 | . 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 1535 ⊤wtru 1536 ∃wex 1774 ∈ wcel 2104 {cab 2710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1538 df-ex 1775 df-sb 2061 df-clab 2711 df-clel 2812 |
This theorem is referenced by: elisset 2819 bj-issettruALTV 36816 bj-issetwt 36818 bj-vtoclg1f1 36860 |
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