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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 2139, ax-12 2175. (Revised by GG, 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 4680 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓)) |
4 | dfrex2 3071 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑) | |
5 | bicom1 221 | . . . 4 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)) | |
6 | 5 | con1bid 355 | . . 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 206 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-sn 4632 |
This theorem is referenced by: rexsn 4687 rextpg 4704 iunxsng 5095 frirr 5665 frsn 5776 imasng 6104 naddunif 8730 scshwfzeqfzo 14862 dvdsprmpweqnn 16919 mnd1 18805 grp1 19078 pzriprnglem3 21512 pzriprnglem10 21519 psdmul 22188 cutmax 27983 cutmin 27984 halfcut 28431 elntg2 29015 1loopgrvd0 29537 1egrvtxdg0 29544 nfrgr2v 30301 1vwmgr 30305 elgrplsmsn 33398 grplsmid 33412 ballotlemfc0 34474 ballotlemfcc 34475 bj-restsn 37065 elrnressn 38255 elpaddat 39787 elpadd2at 39789 brfvidRP 43678 mnuunid 44273 iccelpart 47358 zlidlring 48078 lco0 48273 |
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