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Mirrors > Home > ILE Home > Th. List > rexsnsOLD | GIF version |
Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3409 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rexsnsOLD | ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc5 2787 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
2 | 1 | a1i 9 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
3 | df-rex 2312 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑)) | |
4 | velsn 3392 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
5 | 4 | anbi1i 431 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)) |
6 | 5 | exbii 1496 | . . 3 ⊢ (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
7 | 3, 6 | bitri 173 | . 2 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
8 | 2, 7 | syl6rbbr 188 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 [wsbc 2764 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-sbc 2765 df-sn 3381 |
This theorem is referenced by: rexsng 3412 r19.12sn 3436 |
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