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Mirrors > Home > ILE Home > Th. List > cleljust | GIF version |
Description: When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1440 with the class variables in wcel 1439. (Contributed by NM, 28-Jan-2004.) |
Ref | Expression |
---|---|
cleljust | ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1465 | . . 3 ⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦) | |
2 | elequ1 1648 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | equsex 1664 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦) |
4 | 3 | bicomi 131 | 1 ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1427 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-4 1446 ax-13 1450 ax-17 1465 ax-i9 1469 ax-ial 1473 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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