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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 4166 | . 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 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-sbc 3403 df-sn 4126 |
This theorem is referenced by: elsnres 5356 oarec 7529 snec 7697 zornn0g 9210 fpwwe2lem13 9343 elreal 9831 hashge2el2difr 13118 vdwlem6 15528 pmatcollpw3fi1 20412 restsn 20784 snclseqg 21729 ust0 21833 esum2dlem 29481 eulerpartlemgh 29767 eldm3 30905 poimirlem28 32607 heiborlem3 32782 |
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