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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 1509 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | ceqsexv.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | ceqsexv.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | ceqsex 2727 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: ceqsex3v 2731 gencbvex 2735 sbhypf 2738 euxfr2dc 2873 inuni 4088 eqvinop 4173 onm 4331 uniuni 4380 opeliunxp 4602 elvvv 4610 rexiunxp 4689 imai 4903 coi1 5062 abrexco 5668 opabex3d 6027 opabex3 6028 mapsnen 6713 xpsnen 6723 xpcomco 6728 xpassen 6732 |
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