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Mirrors > Home > MPE Home > Th. List > rexsng | Structured version Visualization version GIF version |
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
Ref | Expression |
---|---|
ralsng.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexsng | ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexsns 4164 | . 2 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑) | |
2 | ralsng.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbcieg 3435 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 1, 3 | syl5bb 271 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 [wsbc 3402 {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: rexsn 4170 rexprg 4182 rextpg 4184 iunxsng 4538 frirr 5015 frsn 5112 imasng 5406 scshwfzeqfzo 13423 dvdsprmpweqnn 15427 mnd1 17154 grp1 17345 frgra2v 26526 1vwmgra 26530 ballotlemfc0 29881 ballotlemfcc 29882 bj-restsn 32216 elpaddat 34108 elpadd2at 34110 brfvidRP 36999 iccelpart 39971 1loopgrvd0 40719 1egrvtxdg0 40727 nfrgr2v 41442 1vwmgr 41446 zlidlring 41718 lco0 42010 |
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