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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 1481 with the class variables in wcel 1480. (Contributed by NM, 28-Jan-2004.) |
Ref | Expression |
---|---|
cleljust | ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1506 | . . 3 ⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦) | |
2 | elequ1 1690 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | equsex 1706 | . 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 1468 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-13 1491 ax-17 1506 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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