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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-abbi2dv | Structured version Visualization version GIF version |
Description: Remove dependency on ax-13 2246 from abbi2dv 2742. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-abbi2dv.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-abbi2dv | ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-abbi2dv.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) | |
2 | 1 | alrimiv 1855 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) |
3 | bj-abeq2 32773 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜓)) | |
4 | 2, 3 | sylibr 224 | 1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: bj-abbi1dv 32781 bj-sbab 32784 |
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