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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-abeq1 | Structured version Visualization version GIF version |
Description: Remove dependency on ax-13 2234 from abeq1 2720. Remark: the theorems abeq2i 2722, abeq1i 2723, abeq2d 2721 do not use ax-11 2021 or ax-13 2234. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-abeq1 | ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-abeq2 31961 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | |
2 | eqcom 2617 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ 𝐴 = {𝑥 ∣ 𝜑}) | |
3 | bicom 211 | . . 3 ⊢ ((𝜑 ↔ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) | |
4 | 3 | albii 1737 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
5 | 1, 2, 4 | 3bitr4i 291 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 {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 df-clel 2606 |
This theorem is referenced by: (None) |
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