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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.) Avoid ax-10 2138, ax-12 2172. (Revised by Gino Giotto, 30-Sep-2024.) |
Ref | Expression |
---|---|
ralsng.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexsng | ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsng.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 318 | . . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | ralsng 4678 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓)) |
4 | dfrex2 3074 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑) | |
5 | bicom1 220 | . . . 4 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)) | |
6 | 5 | con1bid 356 | . . 3 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ 𝜓)) |
7 | 4, 6 | bitrid 283 | . 2 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
8 | 3, 7 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {csn 4629 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-sn 4630 |
This theorem is referenced by: rexsn 4687 rextpg 4704 iunxsng 5094 frirr 5654 frsn 5764 imasng 6083 naddunif 8692 scshwfzeqfzo 14777 dvdsprmpweqnn 16818 mnd1 18667 grp1 18930 elntg2 28243 1loopgrvd0 28761 1egrvtxdg0 28768 nfrgr2v 29525 1vwmgr 29529 elgrplsmsn 32500 grplsmid 32514 ballotlemfc0 33491 ballotlemfcc 33492 bj-restsn 35963 elrnressn 37141 elpaddat 38675 elpadd2at 38677 brfvidRP 42439 mnuunid 43036 iccelpart 46101 pzriprnglem3 46807 pzriprnglem10 46814 zlidlring 46826 lco0 47108 |
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