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Mirrors > Home > MPE Home > Th. List > rexsns | Structured version Visualization version GIF version |
Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
Ref | Expression |
---|---|
rexsns | ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velsn 4607 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | 1 | anbi1i 625 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)) |
3 | 2 | exbii 1851 | . 2 ⊢ (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
4 | df-rex 3075 | . 2 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑)) | |
5 | sbc5 3772 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3074 [wsbc 3744 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rex 3075 df-v 3450 df-sbc 3745 df-sn 4592 |
This theorem is referenced by: rexsngf 4636 r19.12sn 4686 poimirlem25 36132 |
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