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Mirrors > Home > ILE Home > Th. List > axext3 | GIF version |
Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
axext3 | ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ2 1598 | . . . . 5 ⊢ (w = x → (z ∈ w ↔ z ∈ x)) | |
2 | 1 | bibi1d 222 | . . . 4 ⊢ (w = x → ((z ∈ w ↔ z ∈ y) ↔ (z ∈ x ↔ z ∈ y))) |
3 | 2 | albidv 1702 | . . 3 ⊢ (w = x → (∀z(z ∈ w ↔ z ∈ y) ↔ ∀z(z ∈ x ↔ z ∈ y))) |
4 | equequ1 1595 | . . 3 ⊢ (w = x → (w = y ↔ x = y)) | |
5 | 3, 4 | imbi12d 223 | . 2 ⊢ (w = x → ((∀z(z ∈ w ↔ z ∈ y) → w = y) ↔ (∀z(z ∈ x ↔ z ∈ y) → x = y))) |
6 | ax-ext 2019 | . 2 ⊢ (∀z(z ∈ w ↔ z ∈ y) → w = y) | |
7 | 5, 6 | chvarv 1809 | 1 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 |
This theorem is referenced by: axext4 2021 |
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