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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-abbid | Structured version Visualization version GIF version |
Description: Remove dependency on ax-13 2234 from abbid 2727. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-abbid.1 | ⊢ Ⅎ𝑥𝜑 |
bj-abbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bj-abbid | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-abbid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | bj-abbid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimi 2069 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | bj-abbi 31963 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) ↔ {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | |
5 | 3, 4 | sylib 207 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 Ⅎwnf 1699 {cab 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 |
This theorem is referenced by: bj-abbidv 31967 |
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