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Mirrors > Home > ILE Home > Th. List > ceqsexv | GIF version |
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) |
Ref | Expression |
---|---|
ceqsexv.1 | ⊢ 𝐴 ∈ V |
ceqsexv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsexv | ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | ceqsexv.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | ceqsexv.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | ceqsex 2592 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 |
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-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: ceqsex3v 2596 gencbvex 2600 sbhypf 2603 euxfr2dc 2726 inuni 3909 eqvinop 3980 onm 4138 uniuni 4183 opeliunxp 4395 elvvv 4403 rexiunxp 4478 imai 4681 coi1 4836 abrexco 5398 opabex3d 5748 opabex3 5749 xpsnen 6295 xpcomco 6300 xpassen 6304 |
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