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Mirrors > Home > MPE Home > Th. List > rexsn | Structured version Visualization version GIF version |
Description: Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Ref | Expression |
---|---|
ralsn.1 | ⊢ 𝐴 ∈ V |
ralsn.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexsn | ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ralsn.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | rexsng 4614 | . 2 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-sbc 3773 df-sn 4568 |
This theorem is referenced by: elsnres 5892 oarec 8188 snec 8360 zornn0g 9927 fpwwe2lem13 10064 elreal 10553 hashge2el2difr 13840 vdwlem6 16322 pmatcollpw3fi1 21396 restsn 21778 snclseqg 22724 ust0 22828 esum2dlem 31351 eulerpartlemgh 31636 eldm3 32997 poimirlem28 34935 heiborlem3 35106 |
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