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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 4414 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | 1 | anbi1i 619 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)) |
3 | 2 | exbii 1949 | . 2 ⊢ (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
4 | df-rex 3124 | . 2 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑)) | |
5 | sbc5 3688 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
6 | 3, 4, 5 | 3bitr4i 295 | 1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ∃wrex 3119 [wsbc 3663 {csn 4398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rex 3124 df-v 3417 df-sbc 3664 df-sn 4399 |
This theorem is referenced by: rexsng 4440 r19.12sn 4474 iunxsngf 4825 poimirlem25 33979 rexsngf 40038 |
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